Hilbert's third problem and a conjecture of Goncharov
aa r X i v : . [ m a t h . K T ] S e p HILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV
JONATHAN A. CAMPBELLINNA ZAKHAREVICH
Abstract.
In this paper we reduce the generalized Hilbert’s third problem about Dehn invariants andscissors congruence classes to the injectivity of certain Cheeger–Chern–Simons invariants. We also establisha version of a conjecture of Goncharov relating scissors congruence groups of polytopes and the algebraic K -theory of C . We prove, in particular, that the homology of the “Dehn complex” of Goncharov splits asa summand of the twisted homology of a Lie group made discrete. Introduction
Hilbert’s third problem asks the following question: given two polyhedra P and Q , when is it possible todecompose P into finitely many polyhedra and form Q out of the pieces? More formally, is it possible to write P = S ni =1 P i and Q = S ni =1 Q i such that P i ∼ = Q i for all i , and such that meas( P i ∩ P j ) = meas( Q i ∩ Q j ) = 0for all i = j ? (Two polyhedra for this this is true are called scissors congruent .) The generalized version ofHilbert’s third problem is the observation that this can be asked in any dimension and any geometry. Thequestion then becomes: describe a complete set of invariants of scissors congruence classes of polytopes in agiven dimension and geometry.Let us briefly consider the classical version. If two polyhedra are scissors congruent then their volumesare equal. The reverse implication is not true; a second invariant, called the Dehn invariant , exists. Forthree-dimensional polyhedra (in Euclidean, spherical or hyperbolic space) this invariant is defined as follows: D ( P ) = X e edge of P len( e ) ⊗ θ ( e ) ∈ R ⊗ R /π Z . Here, θ ( e ) is the dihedral angle at e ; in other words, it is the arc length of the intersection with P of a smallcircle around e . In dimension n it is possible to define other Dehn invariants, by picking a dimension ℓ andwriting a similar sum over all faces of P of dimension ℓ ; the measure of the angle will then be a portionof the sphere in dimension n − ℓ −
1. By the Dehn–Sydler theorem [Syd65, Jes68] in Euclidean space indimensions 3 and 4, two polytopes are scissors congruent if and only if their volumes and Dehn invariants areequal. Work of Dupont and Sah [DS82] extended this technique to3-dimensional spherical and hyperbolicspace, classifying the kernel of the Dehn invariant as a group homology group. We thus have the followingquestion:
Generalized Hilbert’s Third Problem ([DS82, Question 1]) . In Euclidean, spherical, and hyperbolicgeometries, do the volume and generalized Dehn invariant separate the scissors congruence classes of poly-topes?Remark.
Spherical polytopes are often used to measure angles. When a polytope is decomposed into smallerpolytopes it produces extra angles, at all of the faces along the cuts. These newly produced angles alwaysadd up either to an entire sphere (if they are contained in the interior of the original polytope) or to sometype of “flat” angle. Such flatness can be quantified by observing that such angles always arise from anangle in a lower-dimensional sphere; in spherical scissors congruence classes we thus declare all such anglesto have “scissors congruence measure 0.” The Dehn invariant, and thus the generalized version of Hilbert’sthird problem, can also be defined for polytopes up to such “measure-0” polytopes. See Definition 1.13.An algebraic approach to Hilbert’s Third Problem defines scissors congruence groups , which are freeabelian groups generated by polytopes (in whichever geometry is under consideration) modulo “cutting”and translation by isometries. (For a more formal definition, see Definition 1.12.) Both volume and thegeneralized Dehn invariant can then be defined as homomorphisms of groups, and we see that the generalized
Hilbert’s Third Problem has a positive answer exactly when volume is injective when restricted to the kernelof the Dehn invariant.Motivated by the theory of mixed Tate motives, in [Gon99, Conjecture 1.7] Goncharov proposed thefollowing method for solving generalized Hilbert’s third problem. Let D : P ( S n − ) L ni =1 P ( S i − ) ⊗P ( S n − i ) − ) be the Dehn invariant on the (reduced) spherical scissors congruence groups. The generalizedHilbert’s third problem can then be rephrased to say that volume is injective when restricted to ker D .Conjectures of Ramakrishnan [ ? , Conjectures 7.1.2,7.1.8] imply that the Borel regulator produces an injectivehomomorphism (gr γn K n − ( C ) Q ⊗ ( Q σ ) ⊗ n ) + R / (2 π ) n Q . Here, Q σ is Q with an action of Z / γn K n − ( C ) is the n -th graded piece of the weightfiltration on the algebraic K -theory of C (with action by Z / · + denotes taking the fixed points of the action. (For a more detailed explanation of this, see Section 5; anin-depth understanding of the terms is not needed for the current discussion.) If it were possible to constructan injective map(0.1) ker D ⊗ Q (gr γn K n − ( C ) Q ⊗ ( Q σ ) ⊗ n ) + such that the composition with the Borel regulator was equal to the volume, this would imply generalizedHilbert’s third problem for spherical scissors congruence groups (at least modulo torsion). Goncharov alsomade an analogous form of this conjecture for the hyperbolic groups; here the Borel regulator takes valuesin R . In his paper, Goncharov was able to construct a map of the form (0.1) once polytopes were restrictedto polytopes with algebraic vertices and C was replaced by Q ; however, he did not show that it is injective.There are a couple of indications this initial version may not be the most useful form of the conjecture.Restricting to polytopes of a particular dimension restricts us to considering group homology for matricesof a set dimension; this is directly related to the rank filtration, rather than the γ -filtration. As the gradedpieces of the rank and γ -filtrations are expected to be isomorphic (see, for example, [Knu12, Conjecture 2.6.1]for an in-depth discussion) this is not a major change to the conjecture. Since K ∗ ( C ) Q is isomorphic to theprimitive elements in the Hopf algebra H ∗ ( BGL ( C ); Q ) (where GL ( C ) is considered as a discrete group), thedesired map (0.1) can be described as a map into a quotient of certain group homology groups. The secondis the observation that scissors congruence groups are constructed out of the group homology of orthogonalgroups, rather than general linear groups, so it is more likely that the kernel of the Dehn invariant will berelated to the group homology of orthogonal groups, rather than general linear groups. It turns out thatthe correct analog of the quotient in the orthogonal case is simply the groups H ∗ ( O ( n ; R ); Z [ ] σ ), where · σ indicates that the group is acting via multiplication by the determinant. These also have a regulator, usuallyreferred to as the Cheeger–Chern–Simons class, which agrees with the Beilinson (and thus Borel) regulator[DHZ00] and which is also expected to be injective.With these two changes we can prove Goncharov’s conjecture: Theorem A (Theorem 5.4) . Let D be the Dehn invariant for reduced spherical scissors congruence indimension d . There is an injection ker D H d ( O ( d + 1; R ) , Z [ ] σ ) which, after composition with the Cheeger–Chern–Simons class is equal to the volume map (which is well-defined on ker D ). Thus, if the Cheeger–Chern–Simons class is injective, volume and the Dehn invariantseparate scissors congruence classes in spherical geometry in all dimensions.Analogously, for hyperbolic scissors congruence there is an injection ker D H d ( O (1 , d ; R ) , Z [ ] σ ) which, after composition with the Cheeger–Chern–Simons class is equal to the volume map. Thus if theCheeger–Chern–Simons class is injective volume and the Dehn invariant separate scissors congruence classesin hyperbolic geometry in all dimensions.Remark. For unreduced spherical scissors congruence groups, it is already known that volume and Dehninvariant separate scissors congruence classes [Sah79, Proposition 6.3.22]. However, unreduced sphericalscissors congruence groups do not appear to have nearly as many interesting applications as the reducedversion, and we therefore focus on the reduced case. In the reduced case, the even-dimensional reducedscissors congruence groups are known to be 0 (see Proposition 1.23), as is the homology group in the
ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 3 codomain, so the theorem is vacuous in these cases; however, we state it in full generality to make theanalogy with the hyperbolic case clear.Goncharov’s intuition about scissors congruence classes did not stop at the kernel of the Dehn invariant.He noticed that Dehn invariants can be iterated to produce a chain complex, denoted P ∗ ( S d ). (See Section 3for more details.) In [Gon99], he conjectures [Gon99, Conjecture 1.8] that there exists a homomorphism H m ( P ∗ ( S n − ) ⊗ Q ) (gr γn K n + m ( C ) Q ⊗ ( Q σ ) n ) + for all m . The techniques for proving Theorem A extend to proving a form of this conjecture, as well: Theorem B (Theorem 4.7) . Let X = S d or H d , and let I ( X ) be the isometry group of X . For all m thereare injective homomorphims H m ( P ∗ ( X )) H m + ⌊ d − ⌋ ( I ( X ) , Z [ ] σ ) . In fact, Theorem 4.7 is shown for any field of characteristic 0, not just R ; for the definition of scissorscongruence groups over a general field see Definition 1.21.There main tool allowing us to prove these theorems is the “geometrization” of the Dehn invariant: atopological model which is both rigid and equivariant with respect to the isometry group of our geometry .It is rigid in the sense that the structural properties that we desire of the Dehn invariant (described at thebeginning of Section 2) already hold for the topological spaces, without having to work “up to homotopy”or “inside homology groups.” It is equivariant in the sense that the Dehn invariant is a map of I ( X )-spaces,rather than simply topological spaces.The advantage of this construction is that the presence of higher homological information in the coinvariantcomputations leads to major cancellations. All of the complexity of P ∗ ( X ) is contracted into Q σ . Here,the key observation is that in a topological context homotopy coinvariants and the “total complex” thatGoncharov uses to define P ∗ commute past one another; thus the rigid and equivariant construction of theDehn invariant above can be used to explicitly determine the homotopy type of a space modeling this complex.We produce a spectral sequence whose lowest nonzero row is the complex P ∗ ( S n − ) (resp. P ∗ ( H n − )); thecancellations allowing us to identify the homotopy type of the “total complex” allows us to directly relatethis to the homology of O (2 n ).At the end of our analysis we illustrate the connection between our reformulation and Goncharov’s originalconjectures (see Proposition 5.9). Outline of the Proofs of Theorem B.
The key ingredient in this proof is the repeated use of the notionsof homotopy cofiber and homotopy coinvariants.The Dehn complex is defined as the total complex of a cubical diagram in
AbGp (Definition 3.6), eachvertex of which is obtained by taking coinvariants (i.e. H ) of an action on a Steinberg module. The totalcomplex of a cube is the the same as the total homotopy cofiber taken in the category of chain complexes(Example 3.4). Thus, to construct the Dehn complex, one takes homology of a group, and then takes ahomotopy colimit. This order feels unnatural from the point of view of homotopy theory, as one should firstconstruct the space and then analyze its homology.We begin by replacing each Steinberg module with a space (Definition 1.8), and taking homotopy coinvari-ants of the group action. The key step is the construction of an equivariant Dehn invariant (Definition 2.12),so that we can analyze is the total homotopy cofiber of the original cubical diagram, prior to taking coinvari-ants. We can then commute the homotopy cofiber past the group action, and analyze them independently.We denote this space ( Y X ) hI ( X ) (it is defined in Section 4). Sections 1 and 2 are devoted the constructionof this cube.In the homotopical analysis of ( Y X ) hI ( X ) a minor miracle occurs: the space is weakly equivalent tothe homotopy coinvariants of a sphere with I ( X ) acting on it (almost) trivially. We offer two proofs ofthis fact in Section 6. This allows for significant simplification of the spectral sequences that compute itshomology groups. The spectral sequences that we use are known as the homotopy orbit spectral sequence(Proposition B.7) and the spectral sequence for the total homotopy cofiber of a cube (Section B.3). Ananalysis of the total homotopy cofiber spectral sequence (Lemma 4.5) allows us to immediately concludethat the Dehn complex splits off of H ∗ (( Y X ) hI ( X ) ). We can also use the homotopy orbit spectral sequenceto compute H ∗ (( Y X ) hI ( X ) ) to obtain a shift of H ∗ ( I ( X ); Q σ ); this implies Theorem B. Theorem 5.4 togetherwith Theorem B imply Theorem A. CAMPBELL ZAKHAREVICH
Remark.
In this paper we mostly focus on spherical and hyperbolic geometries, as well as work over R and C , as these were our main examples of interest. However, most of our techniques do not rely on either thesechoices of geometry or the choice of field. In future work we hope to work out further implications of theseapproaches in other fields, geometries, and isometry groups. (The Euclidean case is an obvious candidate.) Remark.
What is especially striking about our approach is that most of the topological spaces we workwith turn out to be homotopy-equivalent to bouquets of spheres. This means, in essence, that they are combinatorial objects, rather than topological. Despite this, the topological approach appears to producesignificantly simpler proofs, and stronger results, than a purely algebraic one.
Organization.
In Section 1 we introduce the basic objects of interest. Although many of the objects anddefinitions are standard, several key definitions (esp. RT-buildings) differ subtly from standard. We haveattempted to highlight these differences in the exposition. Section 2 introduces derived Dehn invariants andstates that they agree with the classical definitions; although the comparison between our objects and theclassical objects is interesting (and we believe a good introduction to simplicial techniques) we postpone thedirect comparison to Appendix A, as it is technical and completely disjoint from the main thrust of the paper.Section 3 recalls Goncharov’s definition of the Dehn complex and shows how to construct a “geometrized”model. Section 4 is the main meat of the topological story: it introduces the key theorem (Theorem 4.1)which allows us to directly compare the homology of the Dehn complex to the group homology of orthogonalgroups. Section 5 proves Theorems A and B and explains the connection between Goncharov’s originalconjectures and the form in which they arise in this paper. Section 6 proves Theorem 4.1 and uses the proofto provide the computation for the claim about volume in Theorem A. This section is largely independentof much of the rest of the paper, dealing mostly with the structure of RT-buildings.
Notation and conventions.
We work in the category of pointed topological spaces and simplicial sets.Thus homology is reduced, and all constructions on spaces are pointed. In particular, homotopy G -coinvariants—denoted • hG —are taken in a correctly-pointed manner, so that ∗ hG ≃ ∗ and ( S ) hG ≃ ( BG ) + .Here, • + denotes adding a disjoint basepoint, so that we can think of S as ∗ + . Note the difference withthe unpointed constructions: in the unpointed case ∗ hG ≃ BG . In general, in order to translate from theunpointed case to the pointed case one adds a disjoint basepoint and then works relative to that point. Allgroups in this paper are considered discrete unless explicitly stated otherwise, so that BG is always theEilenberg–Mac Lane space K ( G, f : X Y which is a bijectionon connected components and such that the induced maps f ∗ : π n ( X, x ) π n ( Y, f ( x )) are isomorphismsfor all choices of basepoint x and all n ≥
1. When two spaces (resp. simplicial sets) X and Y are weaklyequivalent, we denote this by X ≃ Y . Two simply-connected spaces are “weakly equivalent after inverting2” if there exists a map f : X Y such that the induced maps f ∗ : π n ( X, x ) ⊗ Z [ ] π n ( Y, f ( x )) ⊗ Z [ ]are isomorphisms for all n ≥
2. We denote this by X ≃ [2] Y .The notation X/Y will refer to the quotient of spaces (resp. simplicial sets): the topological space (resp.simplicial set) given by collapsing all points to Y to a point. The only exception to this notation will be thegroup quotient R / Z (and scalings thereof) and the group Z / X n either n -dimensional hyperbolic ( H n ) or spherical ( S n ) space. In each case, we thinkof X n as sitting inside R n +1 , with subspaces being cut out by subspaces of R n through the origin (whichintersect, respectively, the plane where x n +1 = 1, the hyperboloid − x + x + · · · + x n , and the sphere in anonempty set). When the dimensions is clear from context we write X instead of X n .For any abelian group A , we write A Q def = A ⊗ Q .The field k is always assumed to have characteristic 0. Acknowledgements.
Both authors were supported in part by the IAS Summer Collaborators program.The second author was supported in part by NSF CAREER-1846767.The work in this paper is deeply indebted to Jean-Louis Cathelineau’s work on homological stability oforthogonal groups and scissors congruence [Cat07, Cat04, Cat03], which inspired many of the approaches inthis paper.The authors would like to thank Richard Hain, Max Karoubi, Cary Malkiewich, Daniil Rudenko, JesseSilliman, Craig Westerland, Charles Weibel, and Ilya Zakharevich for their helpfulness and patience withour questions, especially about regulators.
ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 5 RT-buildings and scissors congruence groups
Our main goal in this section is to establish the basic definitions of the objects we will be using, as many ofthese definitions are not (quite) standard. Many small variations on these definitions exist in the literature(see, for example, [Dup01, Chapter 2], [Cat04]), leading to a combinatorial explosion of choices. In ourexperience only the current choices lead to a consistent rigid derived theory. We work over any infinite basefield k of characteristic 0.1.1. RT-buildings.Definition 1.1 (Based on [Cat04, Definition 1.0.3]) . A geometry over k , X , is a vector space equipped withquadratic form ( E, q ) over k , where q is totally nondegenerate, together with its isometry group I ( X ). The dimension of X is dim E −
1. By definition, I ( X ) = I ( E ): the subgroup of GL ( n + 1; k ) which preserves thequadratic form.When we wish to emphasize that a geometry X has dimension n , we write it as X n . Definition 1.2.
The neat geometries are the spherical geometry S n , given by the quadratic form x + · · · + x n ,and the hyperbolic geometry H n , given by the quadratic form − x + x + · · · + x n .When it is not clear from context, we write S nk or H nk to emphasize that the geometries are over k .In later sections, we will often be considering maps of the form X n X a e ⋆ S b . A map of this sort statesthat we fix a type of geometry (spherical or hyperbolic), and both X ’s are of this same type, of dimensions n and a , respectively. Definition 1.3.
For a geometry X = ( E, q ), where q has signature ( n − , n + ), a subspace U of X is a subsetof P ( E ) corresponding to a linear subspace V of E such that the restriction of q to V is totally nondegenerateand such that the signature ( m − , m + ) of q | V has m − = n − .An angular-subspace U of X is a linear subspace V of E such that the restriction of q to V is totallynondegenerate and such that the signature ( m − , m + ) of q | V has m − = 0.For any subspace or angular-subspace U of X we say that U is represented by V . The dimension of U isdim V −
1; if dim U = 0 we refer to E as a point of X .When k contains √−
1, the condition on the signature is vacuous and subspaces and angular-subspacesare equivalent.
Remark . The condition on the signature may appear artificial, but it is necessary in order to model thetypes of subspaces in question. A geometry X of dimension n can be considered to be sitting inside k n +1 asa submanifold. In the case when k is not algebraically closed, a plane of dimension m may not intersect thissubmanifold in a subspace of dimension m −
1, as desired. The condition on the signature ensures that thiswill happen in the cases of interest in this paper.
Remark . Many of the definitions and results in this paper will also work for the Euclidean geometry, aswell as for geometries with signatures other than (0 , n + 1) and (1 , n ). However, there are enough subtletiesand differences between these cases that in this paper we focus exclusively on the spherical and hyperboliccases.The key structure necessary for the program is the presence of an orthogonal complement for any subspaceand the notion of a projection onto the orthogonal complement.
Definition 1.6.
Let U be an i -dimensional subspace of X , represented by a linear subspace V of E . Wedefine the orthogonal complement U ⊥ of U to be the angular-subspace represented by V ⊥ .If U is a subspace and U ′ is an angular-subspace of X , represented by V and V ′ , then we write U ⊥ U ′ if V ⊥ V ′ . We write U ⊕ U ′ for the subspace represented by V ⊕ V ′ . If V ⊥ V ′ we write U ⊥ U ′ instead of U ⊕ U ′ to emphasize this fact.For subspaces U ⊆ U ′ of X , we write pr U ⊥ U ′ def = U ′ ∩ U ⊥ . The isometry group of pr U ⊥ U ′ is taken to be the subgroup of the isometry group of U ′ that fixes U .We will be using the following three properties of subspaces: CAMPBELL ZAKHAREVICH
Lemma 1.7.
Let X n be a neat geometry and let U i be a subspace of X . Then dim U ⊥ = n − i − . For asubspace V containing U , V is uniquely determined by U and U ⊥ ∩ V . In addition, the induced quadraticform on pr U ⊥ V has positive signature. The key object of study in this paper is the
RT-building associated to a geometry X . Definition 1.8.
Let X be a geometry of dimension n over k .Let T m • ( X ) be the simplicial set whose i -simplices are sequences U ⊆ · · · ⊆ U i of nonempty subspaces of X of dimension at most m . The j -th face map deletes U j ; the j -th degeneracy repeats U j . The isometrygroup I ( X ) acts on T m • ( X ).We define the RT-building of X to be the pointed simplicial set given by F X • def = T n • ( X ) /T n − • ( X ) , with the inherited I ( X )-action. More explicitly, the non-basepoint i -simplices of F X • are sequences U ⊆· · · ⊆ U i , where each U j is a nonempty subspace of X and U i = X . The face maps and degeneracies work asbefore, with the caveat that if U i − = X then d i sends the simplex U ⊆ · · · ⊆ U i to the basepoint.It turns out that the group e H n ( F S n • ) contains vital information about scissors congruence. In fact, thisis the only nonzero homology group of this space: Proposition 1.9.
For i = dim X , e H n ( F X • ) ∼ = 0 . The fact that all homology groups above dim X are 0 is evident from the fact that all nondegeneratesimplices have length at most n + 1. The fact that all (reduced) homology groups below degree n are also0 is more complicated; one can refer to the Solomon–Tits Theorem [Qui73, Section 2], or use the theorydeveloped in Appendix A. As the proof is technical and not illuminating, we defer it to the appendix.1.2. Classical scissors congruence.
We turn our attention to defining the scissors congruence groups.For scissors congruence to be defined we need a notion of a geometry to work within, as well as a notion of“inside” and “outside” for polytopes; thus we will need to be working inside an ordered field. For now wefix k = R , although most of the machinery developed should work equally well over other ordered fields.The basic building block of a polytope (and thus of a scissors congruence group) is a simplex, which canbe defined as a convex hull. Definition 1.10.
Suppose X is a neat geometry of dimension n .A convex hull of a tuple ( a , . . . , a m ) of points in X is any subset of X represented by a cone over b i (cid:26) m X i =0 c i b i ∈ R n +1 (cid:12)(cid:12)(cid:12)(cid:12) c i ≥ ∀ i (cid:27) for any choice of 0 = b i ∈ a i for all i .An m -simplex in X is the convex hull of a tuple ( a , . . . , a m ) which is not contained in an m − X . An m -polytope in X is a finite union of m -simplices; we make no assumptions of convexityor connectedness. When m = n we omit it from the terminology and refer simply to “simplices in X ” or“polytopes in X .” Remark . When X is hyperbolic, a simplex is uniquely determined by its vertices in the following sense.The hyperboloid − x + x + · · · + x n has two connected components, and we think of X as one of thesecomponents and a point of X as the intersection of the representing line with this component. A tupleof points ( a , . . . , a n ) in X thus defines a tuple of vectors in R n +1 , and thus the positive cone above iswell-defined.When X is spherical, there are 2 n +1 possible choices of “sign” of the representatives b i . Thus a simplexis no longer uniquely defined by its vertices.We can now define the scissors congruence group of X : The term “RT-building” is named after Rognes and Tits, as our objects are “halfway” between Tits’ original objects—whichmust start at a nonempty subspace and end at a proper subspace—and Rognes’ spaces D ( V ), which have simplices whichstart at the trivial subspace and end at the full space. ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 7
Definition 1.12.
Let X be a neat geometry over R , and let G be a subgroup of I ( X ). Then the scissorscongruence group of X relative to G , denoted b P ( X, G ), is the free abelian group generated by polytopes in X modulo the relations • [ P ∪ Q ] = [ P ] + [ Q ] if P ∩ Q is contained in a finite union of m − • [ P ] = [ g · P ] for any g ∈ G . Here g acts on P pointwise; as it is in I ( X ) it takes convex hulls toconvex hulls.When G = I ( X ) we omit it from the notation.The case X = S n is more complicated, as convex hulls are now only well-defined up to a certain equivalencerelation. The following definition will make all of the choices in Definition 1.10 equivalent. Definition 1.13.
For any G -module M , the coinvariants of G acting on M are defined to be the group M/ ( m − g · m | g ∈ G, m ∈ M ) . This is isomorphic to the zeroth group homology H ( G, M ).Let(1.14) Σ: M V ⊆ R n +1 dim V = n b P ( V ∩ S n , b P ( S n , V ∩ S n to the union of the two simplices defined by the choiceof representatives in V ⊥ . Denote by P ( S n , G ) the cokernel of the induced mapΣ: H (cid:18) G, M V ⊆ R n +1 dim V = n b P ( V ∩ S n , (cid:19) H ( G, b P ( S n , . When X = S n we define P ( X, G ) def = b P ( X, G ).The cokernel of Σ turns out to be the more “correct” notion of scissors congruence of the sphere, as it ismost often used to measure angles. A subdivision of a polytope adds many angle measures that add up tothe entire sphere; thus, in order to make our definitions treat subdivisions correctly, the entire sphere shouldbe considered to be zero. When we discuss the Dehn invariant in Section 2 this will become clearer, as Dehninvariants are only well-defined inside these reduced scissors congruence groups.
Remark . The notation we are using is somewhat nonstandard. The group b P ( X, G ) is usually denoted P ( X, G ), and the group P ( S n , G ) is generally denoted e P ( S n , G ). In this paper, however, the group of interestis P ( S n , G ) for X = S n , and we would like to unify the notation so that this group is the default one.As motivation for considering scissors congruence as homotopy coinvariants we observe that(1.16) P ( X, G ) ∼ = H ( G, P ( X, b P ( S n , G ) as well.1.3. Geometrizing a twist.
In the classical literature on scissors congruence, flags often take the placeof polytopes (thus motivating our study of RT-buildings). However, a complication arises: in the algebraicstory, the action of the isometry on flags is twisted by the determinant map, so that g · [ x ] = (det g )[ g · x ].In order to construct a topological model of such a twist, we need a topological model for “tensoring with acopy of Z with the sign action.” Definition 1.17.
Let S be the pointed simplicial set ∆ /∂ ∆ .Let S σ be the pointed simplicial set ∆ ∪ ∂ ∆ ∆ , with one of the vertices in ∂ ∆ taken to be the basepoint.This is a model of a circle with two 0-simplices and two 1-simplices. There is an action of Z / S σ givenby swapping the two 1-simplices. Remark . The notation S σ is chosen to be compatible with the standard notation M σ for a G -modulewhich is twisted by the action of a “sign” map G Z / CAMPBELL ZAKHAREVICH
The group I ( X ) acts on S σ via the map det: I ( X ) Z /
2. Note that H n ( F X • ) ∼ = H n +1 ( S σ ∧ F X • ) asgroups. As I ( X )-modules, these differ only by the action on S σ , which adds a twist • σ by the determinant.In particular, this means that H ( G, H n ( F X • ) σ ) ∼ = H ( G, H n +1 ( S σ ∧ F X • )) . In other words, the I ( X )-coinvariants of H n +1 ( S σ ∧ F X • ) are exactly the “ I ( X )-semi-coinvariants” in H n ( F X • ).From the homotopy orbit spectral sequence (see Proposition B.7), we have H ( G, H n +1 ( S σ ∧ F X • )) ∼ = H n +1 (( S σ ∧ F X • ) hG ) . Remark . It may seem that the approach of “geometrizing” • σ by smashing with S σ produces a spuriousincrease of dimension. However, this increase is present the algebraic story (discussed in [Sah79], [Gon99], andothers) as well: the scissors congruence groups for S n − ⊆ R n must be graded by the ambient dimension n ,rather than n −
1, in order to make the Dehn invariant a graded homomorphism. In addition, Section 5 showsthat these dimensions allow the maps from K -theory to have the correct grading. It is thus unsurprisingthat something of this sort should appear in the topological viewpoint.1.4. The geometrization of scissors congruence groups.
We now state the connection between scissorscongruence and RT-buildings:
Theorem 1.20.
Suppose k = R . Let X have dimension n and let G be a subgroup of the isometry group of X . For a neat geometry X , P ( X, G ) ∼ = H n +1 (( S σ ∧ F X • ) hG ) ∼ = H ( G, H n +1 ( S σ ∧ F X • )) . This map is induced by the G -equivariant map P ( X, H n +1 ( S σ ∧ F X • ) which takes a simplex withvertices { x , . . . , x n } to the sum X σ ∈ Σ n +1 sgn( σ )[span( x σ (0) ) ⊆ span( x σ (0) , x σ (1) ) ⊆ · · · ⊆ span( x σ (0) , . . . , x σ ( n ) )] . As the proof of this theorem is technical and not illuminating, we postpone it until Appendix A. Asimilarly-simple model for b P ( S n , G ) is not known.The theorem above implies that scissors congruence information is contained inside H n +1 ( S σ ∧ F X • ) ∼ = H n ( F X • ) σ . The value added by the topology is that when G = 1 the space ( S σ ∧ F X • ) hG contains nontrivialhigher homological information, and thus remembers more about the algebra of G than the left-hand side.Inspired by Theorem 1.20 we can now define generalized scissors congruence groups: Definition 1.21.
Let X be a geometry over k , and let G ≤ I ( X ). The scissors congruence group of X ,written P ( X, G ), is P ( X, G ) def = H n +1 (( S σ ∧ F X • ) hG ) . When G = I ( X ) we omit it from the notation. Remark . At this point it may be tempting to think that since all spaces under consideration are simply-connected, the current topological model can contain no information that is not contained in P ( X, G ).However, this misses the important point that we are keeping track not only of the group, but also ofthe G -action. Taking orbits on the level of homology is the “underived” model, which cannot keep trackof higher homotopical information. Taking homotopy coinvariants will remember this higher information,analogously to the way that taking homotopy coinvariants of G acting on a point produces BG , which hasmany interesting homology groups (even though a point does not).A large part of the value in these models is that we can define a rigid topological model of the Dehninvariant. This allows us to use simple topological techniques to prove Theorem 4.1, none of which work inthe case where we are working up to homotopy.As a first application of the topological viewpoint, we show that when X = S n there is no interestingscissors congruence information: Proposition 1.23.
When n ≥ , after inverting , ( S σ ∧ F S n • ) hO (2 n +1) ≃ [2] ∗ . ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 9
In other words, the left-hand side is connected and for i ≥ , π i (( S σ ∧ F S n • ) hO (2 n +1) ) ⊗ Z [ ] ∼ = H i (cid:16) ( S σ ∧ F S n • ) hO (2 n +1) ; Z [ ] (cid:17) ∼ = 0 . In particular, for n > , P ( S n ) = 0 .Proof. First consider the case when n >
0. The matrix − I ∈ O (2 n + 1) acts on all homology groupsof S σ ∧ F S n • by −
1. Thus by the “Center Kills Lemma” [Dup01, Lemma 5.4], 2 annihilates H i ( O (2 n +1) , e H n ( S σ ∧ F S n • ; Z [ ])) for all i ; since we have inverted 2, these must all be 0. By the homotopy orbitspectral sequence (Proposition B.7), e H ∗ (cid:16) ( S σ ∧ F S n • ) hO (2 n +1) ; Z [ ] (cid:17) = 0for all i . Since the simplicial set ( S σ ∧ F S n • ) hO (2 n +1) is simply-connected (as suspensions and homotopycoinvariants commute), it must be contractible after inverting 2.The last part of the proposition follows because by [Dup01, Corollary 2.5], P ( S n R ) is 2-divisible, soinverting 2 does not affect the lowest homology group.When n = 0 the situation is somewhat more complicated. In this case, F S • = S , with one non-basepoint0-simplex represented by the subspace S and no other nondegenerate simplices. Then S σ ∧ S = S σ , with O (1) = Z / S σ ) h Z / ∼ = B Z /
2, which is a nilpotentspace. Thus after inverting 2 its homotopy groups are trivial, as desired.The final statement in the proposition is a direct consequence of [Sah79, Proposition 6.2.2]: since P ( S n )is 2-divisible inverting 2 does not affect the scissors congruence group and the original group itself must be0. (cid:3) In general, the homology of ( S σ ∧ F X • ) hG can be described in terms of the homology of G and the homologyof F X • : Lemma 1.24.
Let X be a neat geometry of dimension n . For all i , e H i (cid:0) ( S σ ∧ F X • ) hG ) ∼ = H i − ( n +1) (cid:0) G, e H n ( F X • ) σ (cid:1) , where · σ denotes that the action of G on the homology is twisted by multiplication by the determinant. Inparticular, if i − ( n + 1) is negative the left-hand side is .Proof. This follows directly from the homotopy orbit spectral sequence (see Proposition B.7). Since thereduced homology of S σ ∧ F X • is concentrated in degree n + 1, the homotopy orbit spectral sequence iscontained in the n -th column, and thus collapses. This implies that e H i (( S σ ∧ F X • ) hG ) ∼ = H i − ( n +1) ( G, e H n +1 ( S σ ∧ F X • )) . Using that e H n +1 ( S σ ∧ F X • ) ∼ = e H n ( F X • ) σ as a G -module completes the proof. (cid:3) Rigid derived Dehn invariants
The statement (rephrased in modern terminology) of Hilbert’s third problem is extremely simple:Do there exist two polyhedra with the same volume which are not scissors congruent?The answer, given in 1901 by Dehn is “yes”: the cube and regular tetrahedron are not scissors congruent,even if they have the same volume. Dehn proved this statement by constructing a second invariant ofpolyhedra (these days called the “Dehn invariant”) which is zero on a cube and nonzero on any regulartetrahedron. This invariant takes values in R ⊗ Z R / Z —a difficult group to work in, but even more startlinggiven that tensor products were only originally defined in 1938. In 1965, Sydler proved that the volume andthe Dehn invariant uniquely determine scissors congruence classes; phrased in a more modern fashion (after[Jes68]), this is equivalent to stating that the volume map is injective when restricted to the kernel of theDehn invariant. The classical story: constructing an equivariant Dehn invariant.
In this section we give adefinition of the classical Dehn invariant (extended to arbitrary dimensions in the form proposed by Sah in[Sah79]) and construct a derived model (a “geometrization”). The homological inspiration for our construc-tion is Cathelineau’s approach in [Cat03, Cat04, Cat07].
Definition 2.1.
Let X n be a neat geometry over R , and consider P ( X ). For any integer 0 < i < n , wedefine the i -th classical Dehn invariant in the following manner. Since P ( X ) is generated by simplices, itsuffices to define it on simplices. For a simplex σ in X with vertices { x , . . . , x n } , we define b D i ( σ ) = X J ⊔ J ′ = { ,...,n }| J | = i +1 U =span x J [ x J ] ⊗ [pr U ⊥ ( x J ′ )] ∈ P ( X i , I ( X i )) ⊗ P ( S n − i − , I ( S n − i − )) . Here, x J is the set { x j | j ∈ J } , [ x J ] is the class of the simplex with vertices x J in an isometric copy of X i sitting inside X n , and [pr U ⊥ x J ′ ] is the class in P ( S n − i − , I ( S n − i − )) of the simplex spanned by theprojections of the x J ′ . For a more detailed discussion of this, see [Sah79, Section 6.3].This generalization of the Dehn invariant allows for the following question, the “generalized Hilbert’s thirdproblem”: Question 2.2 (Generalized Hilbert’s third problem) . In a neat geometry X n , is volume injective whenrestricted to the kernel of L ni =1 b D i ?Remark . There is an important subtlety which is often overlooked in the definition of scissors congruencegroups. Although volume is well-defined on P ( H n ) and b P ( S n ), it is not well-defined on P ( S n ): inside P ( S n )we quotient out by lunes , which are polytopes which are “suspensions” of lower-dimensional polytopes. Sincelunes of any volume can be constructed, the volume map on b P ( S n ) does not descend to a well-defined mapout of P ( S n ) for n >
1. (When n = 1 all lunes are semicircles, and thus length is well-defined mod π .)It will turn out, however, that on the kernel of the Dehn invariant volume is well-defined, so the questionis well-stated even for the spherical case. (See Theorem 5.3 for more details.)The Dehn invariant is not well-defined if P ( X ) is replaced by P ( X, x I ] as sitting inside P ( X i , I ( X ) for aslong as possible in order to model D i on an RT-building, it is necessary construct the Dehn invariant as thefunctor of I ( X )-coinvariants applied to an equivariant homomorphism of I ( X )-modules; this motivates thelater geometric construction. To define a map b D U : P ( X, P ( U, ⊗ P ( U ⊥ , σ with vertices { x , . . . , x n } in X , and suppose that U = span( x , . . . , x i ). Let τ be theprojection of { x i +1 , . . . , x n } to U ⊥ . Define b D U ([ x , . . . , x n ]) def = [ x , . . . , x i ] ⊗ [ τ ] . For any simplex { x , . . . , x n } such that there do not exist 0 ≤ j < · · · < j i ≤ n with U = span( x j , . . . , x j i ),define b D U ([ x , . . . , x n ]) def = 0 . Lemma 2.4.
With this definition, P ( X, L b D U M U ⊆ X dim U = i P ( U, ⊗ P ( U ⊥ , is well-defined and I ( X ) -equivariant. After taking I ( X ) -coinvariants this map becomes b D i .Proof. To prove well-definedness it suffices to show that for any simplex σ all but finitely many of the b D U are 0. This is true because there are only finitely many subspaces U which are the span of a subset of { x , . . . , x n } . and also P ( E n ), although E n is not a neat geometry ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 11
The action of I ( X ) on the left is simply an action on tuples. The action on the right is a bit morecomplicated: it acts on the indices of the sum, and acts within each group, as well. However, simplices in U can be thought of as i -simplices in X that happen to be contained in U , on each individual simplex theaction is via acting on each vertex of the simplex; in this way the right-hand side is considered to be sittinginside L U ⊆ X dim U = i P ( X, ⊗ P ( U ⊥ , I ( X )-coinvariants makes this map into b D i . The left-hand side becomes P ( X ). Moreover, I ( X ) identifies all of the summands on the right-hand side, and the stabilizer of anyfixed U is I ( U ) × I ( U ⊥ ); thus the right-hand side is H ( I ( U ) × I ( U ⊥ ) , P ( U, ⊗ P ( U ⊥ , H ( I ( U ) , P ( U, ⊗ H ( I ( U ⊥ ) , P ( U ⊥ , H ( I ( U ) × I ( U ⊥ ) , P ( U, ⊗ P ( U ⊥ , , induced by the cross-product in homology, as the group P ( U,
1) is free (by the Solomon–Tits theorem [Qui73,Section 2]). Thus the right hand side is P ( U ) ⊗ P ( U ⊥ ), as desired.To see that the map is exactly b D i , note that taking the I ( X )-coinvariants adds up the images of allnonzero b D U on a given simplex σ ; this is exactly the definition of b D i . (cid:3) The classical Dehn invariant can be iterated in the following sense. Suppose that i < j ; then the followingsquare commutes:(2.5) P ( X n ) P ( X i ) ⊗ P ( S n − i − ) P ( X j ) ⊗ P ( S n − j − ) P ( X i ) ⊗ P ( S j − i − ) ⊗ P ( S n − j − ) . b D i id ⊗ b D j − i b D i ⊗ id b D j Goncharov uses this observation to construct a chain complex of Dehn invariants which he conjectures isrelated to algebraic K -theory. For a discussion of this, see Section 3.2.2. The derived construction.
The goal is to construct a notion of the Dehn invariant on F X • which willproduce the classical Dehn invariant when k = R , but only after taking coinvariants and homology. Theidea that the Dehn invariant should be constructed in this manner is key to making the analysis in Section 4possible. It is also the only perspective from which it appears to be possible to construct the derived Dehninvariant; the authors attempted many non-equivariant constructions before settling on this approach. Thatthis makes the Dehn invariant concise and clean and exposes its combinatorial nature is a minor miracle.The key idea here is to replace the tensor product of abelian groups with the reduced join of simplicialsets. Remark . Classically, the tensor product would be replaced by the smash product, and choosing insteadthe reduced join may appear to be a perverse choice: the reduced join of simplicial sets is not symmetricin the category of simplicial sets, and using it to model a symmetric structure like the tensor product feelsunnatural. And, as above, the authors spent considerable time on attempts to rework this material using asmash product. Unfortunately (or perhaps incredibly interestingly), it does not seem possible to constructa topological model of the Dehn invariant using a smash product of spaces. (See also Remark 1.19.)An interesting corollary of this is that the constructions in this section are fundamentally unstable . Smashproducts of spaces lift naturally to smash products of spectra, and therefore give some hope that analogousconstructions could be lifted to stable models of scissors congruence (such as those arising from [CZ, Zak17]).Unfortunately, this does not appear to be the case, and the natural questions arise: how stable is the Dehninvariant? What parts of it can be seen stably? And which portions are irredeemably unstable? Definition 2.7.
For pointed simplicial sets X and Y , the reduced join X e ⋆ Y is defined by( X e ⋆ Y ) m = _ i + j = m − X i ∧ Y j . For a simplex ( x, y ) ∈ X i ∧ Y j , the face maps d ℓ are defined to be d m × X i ∧ Y j X i − ∧ Y j when ℓ ≤ i ,and 1 × d ℓ − i − : X i ∧ Y j X i ∧ Y j − otherwise. If i = ℓ = 0 or j = m − − ℓ = 0 then the face map Indeed, it seems that this may not be possible with any symmetric notion of product. takes the simplex to the basepoint. Degeneracies are defined analogously, with the first i + 1 acting on the x -coordinate, and the last m − i − y -coordinate. Note that this structure makes the reducedjoin asymmetric .For those unfamiliar with reduced joins, an introduction and proofs of the most relevant properties of thereduced join are in Section B.1. The most important feature of reduced joins is their relationship to smashproducts; the proof is given in Section B.1: Lemma 2.8 (Lemma B.4) . Let X and Y be pointed simplicial sets. There exists a simplicial weak equivalence S ∧ X ∧ Y X e ⋆ Y. To define Dehn invariants on F X • (Definition 1.8) the first step is, as above, to define a Dehn invariantindexed by a single subspace. Definition 2.9.
Let U be a proper nonempty subspace of X . Define the rigidly derived Dehn invariantrelative to U , D U : F X • F U • e ⋆ F U ⊥ • by D U ( U ⊆ · · · ⊆ U n ) = ( ∗ if ∄ j s.t. U j = U ( U ⊆ · · · ⊆ U j ) ∧ (pr U ⊥ U j +1 ⊆ · · · ⊆ pr U ⊥ U n ) if j = max { i | U i = U } . We call a j as above the U -pivot of U ⊆ · · · ⊆ U n .That D U is compatible with the simplicial structure is direct from the definitions. Observe that if a U -pivot exists then it is strictly less than n , since U = X . Lemma 2.10.
Let U be a proper nonempty subspace of X , and write I ( X, U ) for the subgroup of I ( X ) ofthose elements fixing U . The group I ( X, U ) acts on U ⊥ and D U is I ( X, U ) -equivariant.Proof. The first claim follows from the definition of orthogonal complement. To check equivariants it sufficesto check that for any g ∈ I ( X, U ), the map D U commutes with the action of g . If a U -pivot exists then theaction of g passes to F U ⊥ • , and thus commutes with D U . If no U -pivot exists then this is also true afterapplying g ; since g fixes the basepoint the action of g commutes with D U . (cid:3) This derived Dehn invariant can also be iterated on the nose, analogously to 2.5):
Lemma 2.11.
Let U ( V be proper nonempty subspaces of X . Then the following diagram commutes: F X • F U • e ⋆ F U ⊥ • F V • e ⋆ F V ⊥ • F U • e ⋆ F U ⊥ ∩ V • e ⋆ F V ⊥ • D U id e ⋆ D U ⊥∩ V D U e ⋆ id D V Proof.
Fix any m -simplex U ⊆ · · · ⊆ U m with U -pivot i and V -pivot j . For any subspace W of X , ifpr U ⊥ ( W ) ⊆ V ∩ U ⊥ then we must have W ⊆ V . In particular,((1 e ⋆ D U ⊥ ∩ V ) ◦ D U )( U ⊆ · · · ⊆ U m )= (1 e ⋆ D U ⊥ ∩ V )(( U ⊆ · · · ⊆ U i ) ∧ (pr U ⊥ ( U i +1 ) ⊆ · · · ⊆ pr U ⊥ ( U m )))= ( U ⊆ · · · ⊆ U i ) ∧ (pr U ⊥ ( U i +1 ) ⊆ · · · ⊆ pr U ⊥ ( U j )) ∧ (pr V ⊥ pr U ⊥ ( U j +1 ) ⊆ · · · ⊆ pr V ⊥ pr U ⊥ ( U m ))= ( U ⊆ · · · ⊆ U i ) ∧ (pr U ⊥ ( U i +1 ) ⊆ · · · ⊆ pr U ⊥ ( U j )) ∧ (pr V ⊥ ( U j +1 ) ⊆ · · · ⊆ pr V ⊥ ( U m ))where the last step follows because V ⊥ ⊆ U ⊥ . This is equal to the composition around the bottom, asdesired. (cid:3) Up to this point, the definitions and results can be constructed for the smash product, instead of thereduced join. However, the authors could not find anything analogous to the definition below when e ⋆ isreplaced by ∧ : Definition 2.12.
Let 0 < i < n . Define the dimension- i derived Dehn invariant D i to be the lift in thefollowing diagram: ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 13 _ U ⊆ X dim U = i F U • e ⋆ F U ⊥ • F X • Y U ⊆ X dim U = i F U • e ⋆ F U ⊥ • Q D U D i This is well-defined: every simplex contains at most one space of dimension i , and thus only a singledimension- i component will be nontrivial on it. Lemma 2.13. D i is well-defined and I ( X ) -equivariant. This produces a Dehn invariant for a fixed dimension. Moreover, this Dehn invariant can be put into asquare similar to (2.5). When i < j the diagram(2.14) F X • _ U ⊆ X dim U = i F U • e ⋆ F U ⊥ • _ V ⊆ X dim V = j F V • e ⋆ F V ⊥ • _ U ⊆ V dim U = i dim V = j F U • e ⋆ F U ⊥ ∩ V • e ⋆ F V ⊥ • D i id e ⋆ D j − i D i e ⋆ id D j commutes and is rigidly I ( X )-equivariant.Analogously to Theorem 1.20, this construction is compatible with the classical story; as before the proofis postponed to Appendix A. Theorem 2.15.
Suppose dim X = n and k = R . Then H ( I ( X ); H n +1 ( S σ ∧ D i )) is the classical Dehninvariant. A geometrization of the Dehn complex
Let X be a neat geometry (in the sense of Definition 1.2).In [Gon99] Goncharov considers a complex P ∗ ( X ) constructed out of iterations of the Dehn invariant,and gives several conjectures relating these to algebraic K -theory. These conjectures will be discussed inSection 5; here we focus on the construction of this complex and its geometrization.We begin with an informal outline in the case k = R and dim X = 2 n −
1. Using the square (2.5)the classical Dehn invariant b D i can be iterated by varying over all possible values of 0 < i < n − n − j is even, P ( S j ) = 0 (Proposition 1.23); removing the coordinates where theseappear leaves an ( n − Dehn complex and denote it by P ∗ ( X ). One advantage is that it allowsfor the following rephrasing of the generalized Hilbert’s third problem for reduced spherical and hyperbolicscissors congruence groups: Question 3.1.
Is volume injective on H n − P ∗ ( X n ) ? The goal of this section is to develop a tool for analyzing this complex using total homotopy cofibers ofcubical diagrams; as an additional benefit, a definition of the Dehn complex for arbitrary fields k naturallyemerges.More formally: Definition 3.2.
Let I be the category 0 1. An n -cube in C is a functor I n C . Suppose that C is amodel category. Write e I n for the full subcategory of I n which does not contain the object (1 , . . . , F : I n C be any functor. The total homotopy cofiber cofib th F is the homotopy cofiber of the map h colim F | e I n F (1 , . . . , . For a more in-depth discussion of the total homotopy cofiber, see [MV15, Section 5.9].The important examples are the following:
Example . In the case n = 1 the cube F becomes a morphism M M ′ of R -modules, which canbe thought of as a morphism of chain complexes concentrated in degree 0. Taking the homotopy cofiberproduces cofib h ( M [0] M ′ [0]) = (0 M M ′ , with M ′ in degree 0 and M in degree 1. Tautologically, this is the total complex of the 1-complex given bythe original 1-cube. Example . Now consider the general case. Let F : I n Mod R be a functor; this is an n -complex whichhas length 2 in each direction. One can check that the total complex of F is quasi-isomorphic to cofib th F [0].To construct the Dehn complex it is more convenient to work with a slightly different coordinatization ofa cube. Definition 3.5.
Denote by I d the category whose objects are sequences ~A = ( b, a , . . . , a i ) of nonnegativeintegers such that b + a + · · · + a i = d and in which all a j are positive and even. There exists a morphism( b, a , . . . , a i ) ( b ′ , a ′ , . . . , a ′ ℓ ) if there exist indices 0 ≤ i < · · · < i ℓ = i such that b = b ′ + a ′ + · · · + a ′ i and a j = a ′ i j − +1 + · · · + a ′ i j .Note that I d is an ⌊ d − ⌋ -cube via the map ( b, a , . . . , a i ) ( δ , . . . , δ ⌊ d − ⌋ ), where δ j = 0 if there existsan index ℓ with b + a + · · · + a ℓ = 2 j and 0 otherwise. Definition 3.6.
Let X be a neat geometry of dimension d .Define the Dehn complex P ∗ ( X ) to be the total complex (equivalently, total homotopy cofiber) of thecube D : I d AbGp sending ( b, a , . . . , a i ) to D ( b, a , . . . , a i ) = Z [ ] ⊗ P ( X b ) ⊗ i O j =1 P ( S a j − )and the map ( b, a , . . . , a j + a j +1 , . . . , a i ) ( b, a , . . . , a i ) to 1 ⊗ · · · ⊗ D a j ⊗ · · · ⊗ equivariant Dehn cube is the cube D eqvt : I d Mod I ( X ) given by D eqvt ( b, a , . . . , a i ) = Z [ ] ⊗ M W ⊕ ⊥ V ⊕ ⊥ ···⊕ ⊥ V i = X dim W = b dim V j = a j − P ( W, ⊗ i O j =1 P ( V j , . By the same reasoning as in the proof of Lemma 2.4, we see that H ( I ( X ) , − ) ◦ D eqvt = D . Thus the Dehn complex is obtained by constructing a cube of coinvariants of homology groups and takingits total homotopy cofiber. The goal of this section is to construct, I ( X )-equivariantly, a “geometrization”:a cube of spaces that produces D after taking homology and then the I ( X )-coinvariants. Remark . As mentioned in (1.16) and Theorem 1.20, taking coinvariants in the homology can be replacedby taking the homotopy coinvariants of an action on a space. Since homotopy coinvariants and the totalhomotopy cofiber commute past one another, in future sections there are applied apply in the opposite orderto relate the homology of the Dehn complex to algebraic K -theory. Model categories are just one of a wide variety of situations (often called “homotopical categories”) in which it is possibleto define homotopy cofibers (which is all we need for the current application). For an overview of this, see for example [Rie]. The only change that tensoring with Z [ ] imposes is that when n = 0, P ( X n ) ∼ = Z [ ], instead of Z (as it would usually be:it is a count of the number of points). All other classical scissors congruence groups are 2-divisible. ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 15
We proceed as in previous sections: by replacing P ( X ) with F X • . It cannot be done over Z , but insteadover Z [ ]; the difficulties are highlighted in the differences between F ~A • and J ~A • : Definition 3.8.
Let ~A be any tuple of integers ~A = ( b, a , . . . , a i ). Define F ~A • def = _ W ⊕ ⊥ L ⊥ V j = X dim V j = a j − W = b F W • e ⋆ i e ⋆ j =1 F V j • (with e ⋆ -factors ordered from left to right) and J ~A • def = _ W ⊕ ⊥ L ⊥ V j = X dim V j = a j − W = b F W • ∧ i ^ j =1 ( S σ ∧ F V j • ) . The construction of the Dehn complex in spaces can be duplicated in the current context:
Definition 3.9.
Define the functor Y : I d Top by ~A S σ ∧ F ~A • , with morphisms given by the appropriate D i . Define the Dehn space Y X by Y X = cofib th Y . Theorem 3.10.
Let X be a neat geometry of dimension d . The Dehn complex is quasi-isomorphic to thetotal complex of the ⌊ d − ⌋ -cube given by H d +1 (cid:0) Y ( − ) hI ( X ) ; Z [ ] (cid:1) : ~A H d +1 (cid:16) ( S σ ∧ F ~A • ) hI ( X ) ; Z [ ] (cid:17) . The rest of this section is dedicated to the proof of this theorem; as everything from this point on will bedone with Z [ ] coefficients, the coefficients are omitted from the notation. From the homotopy orbit spectralsequence (Proposition B.7) applied to the right-hand side of the given formula, it suffices to construct anatural isomorphism D H ( I ( X ) , H d +1 ( Y )). Because D ∼ = H ( I ( X ) , D eqvt ) it suffices to produce an I ( X )-equivariant natural isomorphism α : D eqvt H d +1 ◦ Y .To produce the value α ~A of α on ~A , first observe that D eqvt ( ~A ) = Z [ ] ⊗ M W ⊕ ⊥ L ⊥ V j = X dim V j = a j − W = b P ( W, ⊗ i O j =1 P ( V j , M W ⊕ ⊥ L ⊥ V j = X dim V j = a j − W = b H b +1 ( S σ ∧ F W • ) ⊗ i O j =1 H a j ( S σ ∧ F V j • ) ∼ = H d +1 (cid:18) _ W ⊕ ⊥ L ⊥ V j = X dim V j = a j − W = b S σ ∧ F W • ∧ i ^ j =1 ( S σ ∧ F V j • ) (cid:19) = H d +1 ( S σ ∧ J ~A • ) . Therefore α ~A could be produced by giving maps of simplicial sets S σ ∧ J ~A • S σ ∧ F ~A • which give isomor-phisms on H d +1 , compatible with the images of arrows in I n . These arrows are given by Dehn invariants;unfortunately, while Definition 2.12 gives a geometrization of the Dehn invariant on F ~A • , we do not have ananalogous geometrization of the Dehn invariant on J ~A • . Therefore the compatibility conditions between the α ~A cannot be stated using only maps of simplicial sets. Instead, ad-hoc mappings are constructed betweenthese spaces, which with a scaling correction behave correctly on homology. These maps are homotopyequivalences after tensoring with Z [ ], although not integral homotopy equivalences. Remark . This seems to imply that the original definition of the Dehn invariant had an extra factor of2 somehow incorporated into the definition. It would be interesting to see a geometric explanation of thisphenomenon.To construct this explicit descriptions of S and S σ are required. The structure is summarized in thefollowing table; note that in the case of S , ǫ = 1 always; in the case of S σ , ǫ = ± S S σ n -simplices {∗ , , . . . , n } {∗ , ⊛ , ± , . . . , ± n } d j ( ǫi ) j = 0 , i = 1 ∗ ⊛ j = i = n ∗ ∗ j < i i − ǫ ( i − i ǫis j ( ǫi ) j < i i + 1 ǫ ( i + 1) j ≥ i i ǫi Z / ǫi
7→ − ǫi All simplices above dimension n are degenerate. All face and degeneracy maps on ∗ (resp. ⊛ ) map it to ∗ (resp. ⊛ ) in the appropriate dimension.In this notation, the simplicial weak equivalence mentioned in Lemma 2.8 is described via(3.13) ( i, x, y ) ∈ ( S ∧ X ∧ Y ) n ( d n − i +1 i x, d i +10 y ) ∈ ( X e ⋆ Y ) n . The key construction for the desired equivalence is the Z / × Z / γ : S σ ∧ S σ S σ ∧ S ( a, b ) ((sgn b ) a, | b | )and γ ( ⋆ ) = ∗ . Here, Z / × Z / Z / × Z / Z /
2. This is a two-fold cover of S by S . More visually, consider the following illustration: ( ∗ , ∗ )( ∗ , ∗ )( ∗ , ∗ ) ( ∗ , ∗ )( ∗ , ⋆ )( ∗ , ⋆ ) ( ⋆, ∗ )( ⋆, ∗ )( ⋆, ⋆ ) (1 , ,
1) ( − , − , , − , − − , − − , − γ ( ∗ , ∗ )( ∗ , ∗ ) ( ⋆, ∗ )( ⋆, ∗ )( ⋆, ∗ ) (2 , , − ,
2) ( − , . In this diagram all nondegenerate simplices present in S ρ × S ρ and S ρ × S are drawn; everything drawndashed is collapsed to a single point in the smash product. Edges are not labeled but 2-simplices are, usingthe explicit description of the simplicial structures in (3.12). In effect, the map γ is the endomorphism ofthe unit disk which multiplies the angle (in polar coordinates) by 2.It is now possible to define f ~A : S σ ∧ J ~A • S σ ∧ F ~A • . For any simplicial sets K and L , let f : S ∧ K ∧ L K e ⋆ L take ( a, x, y ) to ( d n − a +1 a x, d a +10 y ); by Lemma B.4 it is a weak equivalence. Define f ~A inductively,as an i -fold composition of maps of the following form: S σ ∧ K ∧ S σ ∧ L τ S σ ∧ S σ ∧ K ∧ L γ S σ ∧ S ∧ K ∧ L f S σ ∧ K e ⋆ L. Lemma 3.14.
After inverting , f ~A becomes an I ( X ) -equivariant weak equivalence. In an ideal world, it would be possible to define α ~A = H d +1 ( f ~A ). Unfortunately, it is not that simple, asthis is not compatible with Dehn invariants. Consider a small example. When ~A = ( d ), F ~A • = J ~A • = F X • . ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 17
Fix b , and consider the Dehn invariant D b corresponding to the morphism ( d ) ( b, a ). This produces thefollowing noncommutative diagram: H d +1 ( S σ ∧ J ( b,a ) • ) H d +1 ( S σ ∧ F X • ) H d +1 ( S σ ∧ F ( b,a ) • ) b D b H d +1 ( D b ) H d +1 ( f ( b,a )) (cid:8) To make the diagram commute it is necessary to multiply the vertical map by (due to γ having degree2). This is true in general; if | ~A | > f ~A contains | ~A | − γ , and thusmultiplies by 2 | ~A |− in homology. As 2 is inverted, this can be remedied: Lemma 3.15. α ~A def = 2 −| ~A | H d +1 ( f ~A ) gives a natural isomorphism D eqvt H d +1 ( Y ; Z [ ]) . The proof is now complete. (cid:3) Large cubes and the Dehn complex
To construct the Dehn complex, Goncharov essentially starts with the groups P ( X, Y X ) hI ( X ) of derived Dehn invariants defined above. It turns out that the homology of ( Y X ) hI ( X ) can be described in two ways: one by directly analyzing its homotopy type, and one via a spectral sequenceof Munson and Volic [MV15, Proposition 9.6.14]. The comparison between these computations is incrediblyfruitful.For the rest of this section, the neat geometry X over k has dimension d = 2 n or 2 n − n = ⌊ d − ⌋ ). Theorem 4.1.
After inverting there is an equivalence ( Y X ) hI ( X ) ≃ ( S σ ∧ S n − ) hI ( X ) . Here I ( X ) acts by det on the S σ -coordinate and trivially on the S n − -coordinate. The proof of this theorem is deferred to Section 6. The key to this theorem is the observation thathomotopy coinvariants are both homotopy colimits and therefore commute ; thus( Y X ) hI ( X ) = (cofib th Y ) hI ( X ) ≃ cofib th ( Y hI ( X ) ) . This means that the simple combinatorial nature of Y can be played against the benefits of taking homotopycoinvariants. This is also where the benefits of constructing an equivariant Dehn invariant comes into play:if an equivariant model for the Dehn invariant did not exist it would be impossible to move the homotopycoinvariants outside of the total cofiber.
Theorem 4.2.
Write Z [ ] σ for a copy of Z [ ] with I ( X ) acting on it via multiplication by the determinant.For all i , e H i (cid:16) Y XhI ( X ) ; Z [ ] (cid:17) ∼ = H i − n (cid:0) I ( X ); Z [ ] σ (cid:1) ; in particular, when i < n both are zero.Proof. By Theorem 4.1 there is an isomorphism H i ( Y XhI ( X ) ; Z [ ]) ∼ = H i (( S σ ∧ S n − ) hI ( X ) ; Z [ ]). By thehomotopy orbit spectral sequence (Proposition B.7), and since Z [ ] σ ∼ = e H ( S σ ; Z [ ]), e H i (cid:0) ( S σ ∧ S n − ) hI ( X ) ; Z [ ] (cid:1) ∼ = H i − n (cid:16) I ( X ); e H n (cid:0) S σ ∧ S n − ; Z [ ] (cid:1)(cid:17) ∼ = H i − n (cid:0) I ( X ); Z [ ] σ (cid:1) . (cid:3) The spectral sequence for the total homotopy colimit of a cube proved in Proposition B.9 can now beused to to connect the homotopy type of Y XhI ( X ) to the Dehn complex. In this case the spectral sequencebecomes(4.3) E p,q = M ~A =( b,a ,...,a n − − p ) e H q (cid:16) Y ( ~A ) hI ( X ) ; Z [ ] (cid:17) e H p + q (cid:16) Y XhI ( X ) ; Z [ ] (cid:17) . Since Y ( ~A ) has no nonzero homology below degree d + 1 this also holds for Y ( ~A ) hI ( X ) . Thus all entriesin the spectral sequence with q < d + 1 are 0. When q = d + 1 the row of the spectral sequence is exactlythe Dehn complex. In fact, because Y ( ~A ) has homology concentrated in a single degree (Proposition 1.9),a stronger result is possible: Theorem 4.4.
Let X be a neat geometry. Then H ∗ P ∗ ( X d ) is naturally a direct summand of H ∗ + d +1 (( Y X d ) hI ( X d ) ; Z [ ]) . This theorem is a corollary of the following technical result:
Lemma 4.5.
Let G be a discrete group. Let F : I n G Top be an n -cube such that e H ∗ F ( i ) is concentratedin a single degree k > for all i ∈ I n . Let C ∗ be the total complex of H ( G ; e H k F ( · )): I n AbGp . Then H ∗ C ∗ is a direct summand of e H ∗ + k (cofib th F ( · ) hG ) ; moreover, for all i there is a natural projection ǫ i + k : e H i + k (cofib th F ( · ) hG ) H i C ∗ . Definition 4.6.
In the context of the lemma we say that C ∗ is the base complex of F and call ǫ ∗ the projection to the base . Proof of Lemma 4.5.
Since F ( i ) is k − H ( G ; e H k ( F ( i ))) ∼ = e H k ( F ( i ) hG ) . Consider the spectral sequence in Proposition B.9 with E p,q = M P i = n − p e H q ( F ( i ) hG ) e H p + q (cofib th F ( · ) hG ) . Manifestly, the ( q = k )-row of this spectral sequence is C ∗ , with d as the differential. Likewise, e E p,q = M P i = n − p e H q ( F ( i )) e H p + q (cofib th F ) . The natural map F ( i ) F ( i ) hG (induced by 1 G ) gives a map of spectral sequences which is surjectiveat the k -th row (again by the homotopy orbit spectral sequence). Thus to show that all differentials d r in E with r > k -th row vanish, it suffices to show this for e E . But e E is concentrated in the k -th row, hence all d r with r > k -th row of E is a well-defined spectral subsequence; since it is the bottom row of thespectral sequence it is therefore a direct summand of E . Its differential d is by definition the same as C ∗ ;the statement of the lemma follows. (cid:3) We are now ready to prove the theorem.
Proof of Theorem 4.4.
Apply Lemma 4.5 to the group I ( X ) and the functor Y , as each Y ( ~A ) has homologyconcentrated in degree d + 1. This implies that cofib th (cid:16) ~A e H d +1 (cid:16) ( S σ ∧ F ~A • ) hI ( X ) (cid:17)(cid:17) is a direct summandof e H ∗ + d +1 (cofib th Y ( ~A ) hI ( X ) ) = H ∗ + d +1 ( Y XhI ( X ) ). Tensoring up with Z [ ] and applying Theorem 3.10 givescofib th (cid:16) ~A H d +1 (cid:16) ( S σ ∧ F ~A • ) hI ( X ) ; Z [ ] (cid:17)(cid:17) ∼ = P ∗ ( X ) , from which the theorem follows. (cid:3) The upshot of the above two theorems is the following: This is also sometimes called an “edge homomorphism in the spectral sequence.”
ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 19
Theorem 4.7. H i P ∗ ( X ) is a direct summand of H n + i ( I ( X ); Z [ ] σ ) .Proof. Theorem 4.4 proves that H i P ∗ ( X ) splits off of H d +1+ i ( Y XhI ( X ) ; Z [ ]), which by Theorem 4.2 is iso-morphic to H n + i ( I ( X ); Z [ ] σ ). (cid:3) Below is a picture of the spectral sequence for e H ∗ ( Y XI ( X ) ; Z [ ]) in (4.3). The red indicates the non-zeroentries in E . The Dehn complex is the base complex of Y ; it is the thick blue line sitting in the row where q = d + 1.(4.8) d + 1 + n d + 1 d + 1 q pE p,q = M ~A ∈I d len( ~A )= n − p e H q (cid:16) Y ( ~A ) hI ( X ) ; Z [ ] (cid:17) e H p + q − n (cid:0) I ( X ); Z [ ] σ (cid:1) .d : E p,q E p − ,q d r : E rp,q E rp − r,q − r len( b, a , . . . , a i ) = i + 1As it will be needed later in Theorem 5.3, we give an explicit description of a special case of ǫ n − in thisspectral sequence: Lemma 4.9.
Let k = R . In this spectral sequence, the map ǫ n − : H d ( I ( X ); Z [ ] σ ) H n − P ∗ ( X ) is induced by the map taking a chain ( g , . . . , g d ) to the scissors congruence class of the d -simplex with vertices (cid:8) x , g d x , g d g d − x , . . . , g d · · · g x } , (for any chosen point x ∈ X ) with the sign given by Q di =1 det( g i ) . The proof of this lemma is technical and not illuminating, so it is postponed to Section 6 (Lemma 6.6); infact, in that section it is proved over any field k . We state it here for the special case as it makes the resulteasier to describe and this is the only case of interest in the current paper,Directly from the spectral sequence it is possible to prove a minor generalization (to all fields instead ofalgebraically closed ones, and including all d not just odd ones) of a result of Cathelineau. Theorem 4.10. [Cat03, Thm. 10.1.1]
For d = 2 n or n − with n ≥ , and any field k of characteristic , H ( P ∗ ( S d )) ∼ = H n ( O ( d + 1); Z [ ] σ ) H ( P ∗ ( H d )) ∼ = H n ( O (1 , d ) , Z [ ] σ ) . Proof.
Consider the spectral sequence (4.8). In the p + q = d + 1 diagonal there is exactly one nonzero entry: H ( P ∗ ( X d )). (cid:3) Goncharov’s conjectures
In this section we discuss the connections between Goncharov’s original conjectures, Cheeger–Chern–Simons invariants, and the results of the previous sections.In [Gon99], Goncharov has a series of three conjectures about possible connections between the Dehncomplex and the algebraic K -theory of C . We give a summary of these conjectures here. Our notation doesnot exactly agree with Goncharov’s; in particular, Goncharov’s Dehn complex is cohomologically graded and1-indexed, while ours is homologically graded and 0-indexed. We number the parts of our summary by thenumber of the conjecture in [Gon99].All tensor products of Z / Z / Q nσ def = ( Q σ ) ⊗ n , equipped with the diagonal action of Z / Conjecture 5.1 ([Gon99, Conjectures 1.7-1.9]) . Let P ∗ ( X n − ) be the Dehn complex for the geometry X n − over R . (1.8) There exist homomorphisms H i P ∗ ( S n − ) φ i (gr γn K n + i ( C ) Q ⊗ Q nσ ) + and H i P ∗ ( H n − ) φ i (gr γn K n + i ( C ) Q ⊗ Q nσ ) − . (1.7) The homomorphism φ n − is injective, and the diagrams ker n − M i =1 b D i ∼ = H n − P ∗ ( S n − ) (gr γn K n − ( C ) Q ⊗ Q nσ ) + R / (2 π ) n Z φ n − vol r and ker n − M i =1 b D i ∼ = H n − P ∗ ( H n − ) (gr γn K n − ( C ) Q ⊗ Q nσ ) − R φ n − vol r commute. Here, the right-hand map is the Borel (resp. Beilinson) regulator. (1.9) All φ i are isomorphisms.Here, gr γn is the n -th graded part of the γ -filtration, and Q nσ is the vector space Q with Z / acting on it viamultiplication by ( − n . The sign in the superscript indicates taking the ± eigenspace with respect to theaction by complex conjugation. For an exposition of the γ -filtration, see for example [Gra94]. For an exposition of the Borel and Beilinsonregulators see [BG02, Chapter 9].Goncharov proves (1.7) in the case when C is replaced with Q and simplices in the Dehn complex arerestricted to those with algebraic vertices [Gon99, Theorem 1.6]. Note that any polytope which can appearas the fundamental domain of a group action is automatically in the kernel of all Dehn invariants; thus inparticular Goncharov’s conjectures would imply that all volumes of hyperbolic manifolds must be in theimage of the Borel regulator.Inspired by the conjectures, we propose an alternative method to connect the algebraic K -theory of C andthe scissors congruence groups (see Proposition 5.9). Explaining the γ -filtration and the Borel and Beilinsonregulators in the above theorems is extremely nontrivial, while the corresponding notions in our approachare much more elementary.We begin with a description of the Cheeger–Chern–Simons class, which plays the same role as the Borelregulator for the case of orthogonal groups (rather than general linear groups).The construction is originally due to Cheeger and Simons [CS85, Section 8] (although the authors originallylearned it from Dupont [Dup01, Sect. 10]) and works for more general homogeneous spaces. See also [DK90]and [DHZ00, Section 5]. Definition 5.2.
The
Cheeger–Chern–Simons construction is a homomorphism
CCS : H n − ( O (2 n ; R ); Z [ ] σ ) b P ( S n − ) / [ S n − ] , defined as follows. Consider the space O (2 n ; R ) /O (2 n − R ) with the usual topology. This is homeomorphicto S n − with a distinguished point. The group O (2 n ; R ) acts on this on the left, moving the distinguishedpoint. A chain in degree 2 n − g , . . . , g n − ); call such a chain ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 21 generic if the points { x, g n − x, g n − g n − x, . . . , g n − · · · g x } all lie in a single open hemisphere. For ageneric chain, define a geodesic simplex in S n − associated to this chain by∆ ( g ,...,g n − ) def = ( x , g n − x , g n − g n − x , . . . , g n − · · · g x ) . To check that this morphism is well-defined it suffices to check that given any 2 n -chain ( g , . . . , g n ) allof whose faces are generic, the sum over the boundary is 0. This holds in P ( S n − ) / [ S n − ], and thisconstruction can be extended to all of H n − ( O (2 n ; R ); Z [ ] σ ) as the generic chains are dense. Define CCS : H n − ( O (2 n ; R ) δ ; Z [ ] σ ) b P ( S n − ) / [ S n − ][( g , . . . , g n − )] [∆ ( g ,...,g n − ) ](See [CS85, Section 8] and [DK90] for a more in-depth discussion.)Now fix a volume form v ∈ Ω n − ( S n − ) (which we normalize to so that R S n − v S n − = (2 π ) n ). The Cheeger–Chern–Simons class is the homomorphismCCS: H n − ( O (2 n ; R ); Z [ ] σ ) CCS b P ( S n − ) / [ S n − ] vol R / (2 π ) n Z , Here we restricted to the even-dimensional case because (by the “center-kills” lemma [Dup01, Lemma5.4]) the homology groups H n − ( O (2 n ; R )); Z [ ] σ ) are all 0. However, this construction works equally wellfor odd-dimensional groups when this is not the case. By considering H d ∼ = O + (1 , d ; R ) /O ( d ; R ) one obtainsan analogous homomorphism CCS : H d ( O (1 , d ); Z [ ] σ ) P ( H d )and CCS: H d ( O (1 , d ; R ) δ ; Z [ ] σ ) R . A close analysis of this definition gives the following theorem:
Theorem 5.3.
Volume is well-defined on H n − P ∗ ( S n − ) . The Cheeger–Chern–Simons class is the compo-sition of the projection to the base (Definition 4.6) and the volume: CCS: H n − ( O (2 n ; R ) δ ; Z [ ] σ ) ǫ n − H n − P ∗ ( S n − ) vol R / (2 π ) n Z . Analogously, in the hyperbolic case with d = 2 n or n − , it is the composition CCS: H d ( O (1 , d ; R ) δ ; Z [ ] σ ) ǫ n − H n − P ∗ ( H d ) vol R . Proof.
First, consider the hyperbolic case. Volume is well-defined on P ( H d ). The key observation to prove thetheorem is that the group homomorphism CCS agrees with the explicit description of ǫ n − after compositionwith the isomorphism P ( H d ) H d +1 (( S σ ∧ F H d • ) I ( H d ) ; Z [ ]) in Theorem 1.20.Now consider the spherical case. Let p : b P ( S n − ) / [ S n − ] P ( S n − ) be the projection. The keyobservation here is that p ◦ CCS = ǫ n − (by Lemma 4.9). The image of CCS thus lies in p − ( H n − P ∗ ( S n − )),and the natural section of ǫ n − induces a volume mapvol: H n − P ∗ ( S n − ) H n − ( O (2 n ; R ); Z [ ] σ ) CCS R / (2 π ) n Z . The rest of the proof follows as in the hyperbolic case. (cid:3)
Theorem 5.3 implies the following analog of Goncharov’s Conjecture 1.7:
Theorem 5.4.
The inclusion of the direct summand H n − P ∗ ( S n − R ) H n − ( O (2 n ; R ) δ , Z [ ] σ ) fits intoa commutative diagram H n − P ∗ ( S n − R ) H n − ( O (2 n ; R ) δ , Z [ ] σ ) R / (2 π ) n Z . vol CCS2 CAMPBELL ZAKHAREVICH Analogously, the inclusion H n − P ∗ ( H n − R ) H n − ( O (2 n, R ) δ , Z [ ] σ ) fits into a commutative diagram H n − P ∗ ( H n − R ) H n − ( O (2 n − , R ) δ , Z [ ] σ ) R . vol CCS In particular, vol is injective if
CCS is. In fact, if
CCS is injective then the inclusions above are isomor-phisms.Proof.
The first part of the theorem is a simple restatement of Theorem 5.3 using the fact that ǫ is a sectionof the inclusion. If CCS is injective then the diagrams directly imply that vol must be, as well. However,since CCS is defined using a section of ǫ , if it has no kernel then neither does ǫ , which implies that ǫ is anisomorphism, as desired. (cid:3) In order to relate the homology of isometry groups to algebraic K -theory, we introduce the rank filtration. Definition 5.5 ([Wei13, p. 296]) . The higher rational K -theory of k is defined to be K ∗ ( k ) Q def = primitive elements of the Hopf algebra H ∗ GL ( k ) . (Recall that all homology is taken with rational coefficients.) The rank filtration on K ∗ ( k ) Q is defined by F i K ∗ ( k ) Q def = K ∗ ( k ) Q ∩ (cid:0) im ( H ∗ GL ( i ; k ) H ∗ GL ( k )) (cid:1) . Then gr rk n K ∗ ( k ) Q def = F n K ∗ ( k ) Q /F n − K ∗ ( k ) Q . We will also need an auxiliary object; define CL n,m ( k ) def = coker (cid:0) ( H m GL ( n − k )) P ( H m GL ( n ; k )) P (cid:1) , where ( H m GL ( i ; k )) P is the subgroup of H m GL ( i ; k ) of those elements whose images in H ∗ GL ( k ) is primitive.Observe that there is a natural surjection CL n,m ( k ) gr rk n K m ( k ) Q .Analogously to CL n,m we define CO n,m ( k ) def = coker(( H m SO (2 n − k )) P H m SO (2 n ; k ) P ) , where ( H m SO ( i ; k )) P denotes those elements whose image in H ∗ SO ( k ) is primitive. We assume that thestabilization map SO (2 n − k ) SO (2 n ; k ) adds coordinates at positions n and 2 n , rather than at 2 n − n .There is an action of Z / H ∗ SO ( n ; k ) given by conjugation by a matrix with determinant −
1. Fromthe exact sequence SO ( n ; k ) O ( n ; k ) Z / SO ( n ; k ) splits into eigenspaces H m SO ( n ; k ) ∼ = H m O ( n ; k ) ⊕ H m ( O ( n ; k ) , Q σ ) , where the first component is the +1-eigenspace and the second is the − rk n HK ∗ ( k ) Q and CO n,m ( k ). When n is odd H m ( O ( n ; k ); Q σ ) = 0 for all m ([Cat07, Theorem 1.4], or [Dup01, Lemma 5.4]);in particular, this implies that(5.6) CO n,m ( k ) − ∼ = H m ( O (2 n ; k ) , Q σ ) . We now turn our attention to explaining the connection between rational homology of orthogonal groupsand algebraic K -theory. The comparison between the two is very well-studied (see, for example, [BKSOsr15]),and it is known that, after rationalizing, Hermitian K -theory is isomorphic to the homotopy fixed points ofalgebraic K -theory under the action sending a matrix to its transpose [BKSOsr15, (1-c)] via the hyperbolicmap (defined below). This is explored in more detail below, and used to explain the connection betweenGoncharov’s conjectures and the theorems proved above. In particular, we will explain why, in the real case,both spherical and hyperbolic geometries arise, and how Goncharov’s curious twisting factors Q nσ arise. ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 23
Definition 5.7.
The hyperbolic map :hyp: GL ( n ; k ) SO ( n, n ; k ): M (cid:18) M ( M σ ) − (cid:19) . When k contains i = √− √
2, conjugation by the matrix D n def = 1 √ (cid:18) I n I n − iI n iI n (cid:19) det D n = i n induces an isomorphism SO ( n, n ; k ) ∼ = SO (2 n ; k ). Conjugation by the matrix D [1] n def = 1 √ (cid:18) I n I n − diag(1 , i, . . . , i ) diag(1 , i, . . . , i ) (cid:19) det D [1] n = i n − induces an isomorphism SO ( n, n ; k ) ∼ = SO (1 , n − k ).This map is not an isomorphism on homology, and it is known that in the limit as n ∞ it has a largekernel. Definition 5.8.
For any field extension
L/k , the groups H m GL ( n ; k ) and H m SO (2 n ; k ) have an inducedaction by Gal( L/k ). We call this the
Galois action and write it on the left . We call the action inducedby conjugation by a matrix with determinant − H ∗ SO ( n ; k ) the conjugation action , and we write iton the right . These two actions commute, and are equivariant with respect to the stabilization maps, soinduce actions on CL n,m and CO n,m . In the current context only quadratic extensions are considered, and ± − , as before. Thus, for example, the space CO n,m ( k ) + − is thesubspace of those vectors which are − Q σ is considered to have “Galois action” by the sign action, and abuse notation to considerthe “Galois action” on CL m,n ( L ) ⊗ Q nσ via the diagonal action.In the case when a field k does not contain √− k and the algebraic K -theory of k ( i ). The case when k = R is the case of Goncharov’s original conjectures. Proposition 5.9.
Let k be a field containing √ and not containing i = √− . There exist natural zigzags (gr n K m ( k ( i )) Q ⊗ Q nσ ) + ( CL n,m ( k ( i )) ⊗ Q nσ ) + H m − n P ∗ ( S n − k ( i ) ) + H m − n P ∗ ( S n − k ) and (gr n K m ( k ( i )) Q ⊗ Q nσ ) − ( CL n,m ( k ( i )) ⊗ Q nσ ) − H m − n ( P ∗ ( H n − k ( i ) )) + H m − n P ∗ ( H n − k ) . Here, the middle map is induced by the hyperbolic map and projecting to the base. All homology is takenwith rational coefficients.
To make the above analysis as satisfying as possible, it is desirable to prove the following algebraicconjecture:
Conjecture 5.10.
The map H ∗ ( SO (2 n ; k ); Q ) H ∗ ( SO (2 n ; k ( i )); Q ) + is an isomorphism. If this conjecture were true then the zigzags in Corollary 5.9 would be shortened to a length-2 becausethe rightmost inclusions would be isomorphisms.6.
Proof of Theorem 4.1
In this section we prove Theorem 4.1. First, some notation. Write b I d for the category whose objects aresequences ~A = ( b, a , . . . , a i ) of nonnegative integers such that b + a + · · · + a i = d and all a i are positive, andmorphisms defined as in Definition 3.5. Recall the definition of F ~A • in Definition 3.8. Let Φ eqvt : b I n Top ∗ be defined by Φ eqvt ( ~A ) def = S σ ∧ F ~A • and Φ: b I d Top ∗ be defined by Φ( ~A ) = Φ eqvt ( ~A ) hI ( X ) . The functor Φ eqvt is defined the same as the definition of Y in Definition 3.9, extended from I d to b I d . Define Z eqvt def = cofib th Φ eqvt and Z def = cofib th Φ . Then (as total homotopy cofibers and homotopy coinvariants commute)( Z eqvt ) hI ( X ) ≃ Z. Surprisingly, it is possible to identify the homotopy type of Z eqvt . Proposition 6.1.
There is an I ( X ) -equivariant weak equivalence Z eqvt ≃ S σ ∧ S d . Here, the I ( X ) -action is trivial on the S d -coordinate and acting by the determinant on S σ . As the proof of this is technical we postpone it to the end of the section; for now we assume it andcomplete the proof of Theorem 4.1. Note that this proposition is an integral statement; it is not necessaryto invert 2.We now characterize Z hI ( X ) from a different perspective. For ~A = ( b, a , . . . , a i ), by Lemma 3.14, (cid:16) S σ ∧ F ~A • (cid:17) hI ( X ) ≃ [2] ( S σ ∧ F W • ) hI ( X b ) ∧ i ^ j =1 ( S σ ∧ F V j • ) hO ( a j ) . By Proposition 1.23, if any of the a j are odd then ( S σ ∧ F V i • ) hO ( a i ) ≃ [2] ∗ ; thus if ~A has some a j odd thenΦ( ~A ) is contractible. For any atomic morphism ι : ( b, a , . . . , a i ) ( b, a , . . . , a ′ ℓ , a ′′ ℓ , a ℓ +1 , . . . , a i ) (where a ℓ = a ′ ℓ + a ′′ ℓ ) we say that the morphism is in direction r if r = a ′′ ℓ + a ℓ +1 + · · · + a i . If r is odd then Φ( b, a , . . . , a ′ ℓ , a ′′ ℓ , . . . , a i ) hI ( X ) is contractible; thus all morphisms in Φ( b I d ) in odd directionshave contractible codomain. Note that I d is exactly the subcategory of b I d containing all atomic morphismsin even directions.Note that there are ⌊ d − ⌋ even directions and ⌊ d +12 ⌋ odd directions.It is possible to compute total homotopy cofibers iteratively : taking all cofibers in a single direction r ,it produces a cube one dimension lower; the total homotopy cofiber of this cube is equivalent to the totalhomotopy cofiber of the original cube. Take homotopy cofibers in all of the even directions first: this leaves a ⌊ d +12 ⌋ -cube with a single entry Y XhI ( X ) (at the source) and all other entries contractible; since the homotopycofiber of any map X ∗ is Σ X , Z ≃ Σ ⌊ d +12 ⌋ ( Y XhI ( X ) ) . By the homotopy orbit spectral sequence (see Proposition B.7) and Proposition 6.1, H i (cid:0) ( Z eqvt ) hI ( X ) ; Z [ ] (cid:1) ∼ = H i − ( d +1) (cid:0) I ( X ); Z [ ] σ (cid:1) . Thus H i (cid:0) Y hI ( X ) ; Z [ ] (cid:1) ∼ = H ⌊ d +12 ⌋ + i (cid:0) Z ; Z [ ] (cid:1) ∼ = H ⌊ d +12 ⌋ + i (cid:0) ( Z eqvt ) hI ( X ) ; Z [ ] (cid:1) ∼ = H i −⌊ d − ⌋ (cid:0) I ( X ); Z [ ] σ (cid:1) , completing the proof of the theorem. (cid:3) It remains to prove Proposition 6.1.We begin by computing the homotopy cofiber of a single Dehn invariant. In order to be able to do thisfor any general map in the cube, it is necessary to generalize the definition of the Dehn invariant.Let
W, U , . . . , U i be any decomposition of X into orthogonal subspaces. Define d j = dim W + j X ℓ =1 dim U ℓ j ≥ . (Thus d = dim W and d i = dim X .) Let ℓ be an integer distinct from d , . . . , d i . Let j be the minimal indexsuch that d j > ℓ . For convenience, define D ℓ : F W • e ⋆ F U • e ⋆ · · · e ⋆ F U i • _ V j ⊆ U j dim V j = ℓ − d j − F W • e ⋆ F U • e ⋆ · · · e ⋆ F V j • e ⋆ F V ⊥ j ∩ U j • e ⋆ · · · e ⋆ F U i • ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 25 to be 1 e ⋆ · · · e ⋆ D ℓ − d j − e ⋆ · · · e ⋆ Definition 6.2.
For any subset I ⊆ { , . . . , d } let N I F X • be the subspace of F X • containing no subspacewith dimension contained in I .This definition gives a convenient way to identify the total homotopy cofiber of a Dehn cube. Lemma 6.3.
Let X be a pointed simplicial set, and let Y , . . . , Y n be subspaces of X . Write P ( n ) for thepartial order of subsets of { , . . . , n } . Define a functor F : P ( n ) Top ∗ by I X (cid:30) [ i ∈ I Y i , with the induced morphisms given by the quotient maps. Then cofib th F ≃ Σ n n \ i =1 Y i . Proof.
We prove this by induction on n . When n = 0 the cube is trivial and the statement holds. When n = 1 the cube is X X/Y , and the total homotopy cofiber is Σ Y , as desired.Now consider the general case. The total homotopy cofiber can be computed iteratively [MV15, Propo-sition 5.9.3] by first taking cofibers in the direction of “adding n to a set”: the morphisms in which eachsubset J ∈ P ( n −
1) is mapped to J ∪ { n } . Taking the homotopy cofiber for each such J produces the cube G : P ( n − Top ∗ given by J Σ Y n (cid:30) [ j ∈ J Σ( Y j ∩ Y n ) . This is an n − th G ≃ Σ n − n − \ i =1 Σ( Y i ∩ Y n ) . Σ( Y i ∩ Y n ) sits inside Σ X as Σ Y i ∩ Σ Y n ; thenΣ n − n − \ i =1 Σ( Y i ∩ Y n ) = Σ n − n \ i =1 (Σ Y i ) ∩ (Σ Y n ) = Σ n − n \ i =1 Σ Y i = Σ n n \ i =1 Y i , as desired. (cid:3) Proposition 6.4.
Let I ⊆ { , . . . , d } . Consider the sub- | I | -cube formed by D i for i ∈ I and containing theinitial point. This cube has total homotopy cofiber Σ | I | N I F X • .Proof. For conciseness, write D I for the composition of the D i for i ∈ I . Since the Dehn cube commutes,the order of composition is irrelevant. Let I = { i , . . . , i j − } . We claim that D I : F X • _ W ⊕ ⊥ U ⊕ ⊥ ···⊕ ⊥ U j = X dim W = i dim U ℓ = i ℓ − i ℓ − −···− i ℓ Proof of Proposition 6.1. By Proposition 6.4, Z ≃ S σ ∧ Σ d N { , ,...,d − } F X • . However, N { , ,...,d − } F X • ∼ = S ,as it has exactly two simplices in each dimension: the basepoint and X = · · · = X . The functor S σ ∧ · commutes with taking homotopy cofibers; thus Z ≃ S σ ∧ S d . (cid:3) To finish up this section we use this calculation to prove Lemma 4.9. First, a few definitions. Denote by ~g a tuple ( g , . . . , g j ) of elements in I ( X ); this tuple can be of any length j . For 0 ≤ ℓ ≤ j and a coefficient m ∈ Z [ ], define the notation d ℓ ( m~g ) def = m (det g )( g , . . . , g j ) if ℓ = 0 m ( g , . . . , g ℓ +1 g ℓ , . . . , g j ) if 1 ≤ ℓ < jm ( g , . . . , g j − ) if j = ℓ. Write, for 1 ≤ a ≤ b ≤ j ,Π ba ~g def = g b · · · g a and ∐ ba ~g def = g − a · · · g − b = (cid:0) Π ba ~g (cid:1) − . The double complex C ∗∗ is a homologically-graded double complex in which for j ≥ ≤ i < d thegroup C ij is generated by symbols of the form( g , . . . , g j ) { x | · · · | x i } , where g , . . . , g j ∈ I ( X ) and x , . . . , x i ∈ X . When j is clear from context we sometimes write this ~g { x | · · · | x i } . Define boundary maps ∂ h : C ij C ( i − j and ∂ v : C ij C i ( j − by ∂ h ( ~g { x | · · · | x i } ) def = ~g i X ℓ =0 ( − ℓ { x | · · · | b x ℓ | · · · | x i } and ∂ v ( ~g { x | · · · | x i } ) def = ( d ~g ) { g x | · · · | g x i } + j − X ℓ =1 ( − ℓ ( d ℓ ~g ) { x | · · · | x i } . Lemma 6.5. Let σ def = P i m i ~g ( i ) represent a cycle in H d ( I ( X ); Z [ ] σ ) and fix any point x ∈ X . Inside thetotal complex of C ∗∗ the cycle P i m i ~g ( i ) {} is homologous to ( − d +1 X i m i (det Π d ~g ( i ) ) d X ℓ =0 ( − ℓ n x (cid:12)(cid:12)(cid:12) g ( i ) d · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) \ Π dd − ℓ +2 ~g ( i ) · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) Π d ~g ( i ) · x (cid:12)(cid:12)(cid:12) Π d ~g ( i ) · x o , (where the ℓ = 0 term removes the g ( i )1 x ). ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 27 Proof. Fix x ∈ S d . For any ~g = ( g , . . . , g j ), any 1 ≤ λ ≤ j } , and any point y ∈ S d , and any m ∈ Z [ ],define ∆ λ ( m~g, y ) def = m ( g , . . . , g λ ) n y (cid:12)(cid:12)(cid:12) ∐ j ~g · x (cid:12)(cid:12)(cid:12) ∐ j − ~g · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∐ λ ~g · x o ∈ C ( j − λ +2) λ B λ ( m ( a , . . . , a j )) def = m ( g , . . . , g λ ) n ∐ j ~g · x (cid:12)(cid:12)(cid:12) ∐ j − ~g · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∐ λ ~g · x o ∈ C ( j − λ +1) λ . For any ~g = ( g , . . . , g j ) we have ∂ v ∆ λ ( ~g, y ) = ∆ λ − ( d ~g, g y ) + λ − X ℓ =1 ( − ℓ ∆ λ − ( d ℓ ~g, y ) + ( − λ ( g , . . . , g j − ) n y (cid:12)(cid:12)(cid:12) ∐ j ~g · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∐ λ ~g · x o , and ∂ h ∆ λ − ( ~g, y ) = B λ − ( ~g ) + ( − j +1 j X ℓ = λ ( − ℓ ∆ λ − ( d ℓ ~g, y ) + ( − j + λ ( g , . . . , g λ − ) n y (cid:12)(cid:12)(cid:12) ∐ j ~g · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∐ λ ~g · x o . Now consider σ , so that in the formulas above j = d . Define α λ def = X i m i (cid:18) ∆ λ ( d ~g ( i ) , g ( i )1 x ) + d X ℓ =1 ( − ℓ ∆ λ ( d ℓ ~g ( i ) , x ) (cid:19) ∈ C ( d +1 − λ ) λ . The above calculations (which will have j = d − 1) imply that ∂ v α λ = ∂ h α λ − , using the fact that (since σ is a cycle) X i m i j X ℓ =0 ( − ℓ B λ − ( d ℓ ~g ( i ) ) = 0 . Thus, in the total complex, ∂ d X ℓ =1 ( − ℓ α λ = ∂ v α + ( − d ∂ h α d . Plugging in the definitions produces that ∂ h α d = σ and ∂ v α = X i m i (det g ( i )1 ) d X ℓ =0 ( − ℓ n g ( i )1 x (cid:12)(cid:12)(cid:12) ∐ d ~g ( i ) · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) \ ∐ d +1 − ℓ ~g ( i ) · x (cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12) ∐ ~g ( i ) x (cid:12)(cid:12)(cid:12) x o + X i m i d X ℓ =0 ( − ℓ ∆ ( d ℓ ~g ( i ) , x ) . (Here we abuse notation and declare that ∐ d +12 ~g ( x ) = g .) As σ is a cycle, the second sum is 0. Tocomplete the proof observe that for any d + 1-tuple of points ( y , . . . , y d ), the class P dℓ =0 { y | · · · | b y ℓ | · · · | y d } is a horizontal cycle, and is therefore homologous to d X ℓ =0 { g · y | · · · | [ g · y ℓ | · · · | g · y d } for any g ∈ I ( X ). Thus each term in the sum (over i ) for ∂ v α is a cycle. Acting on the i -th term by Q d ~g ( i ) gives the desired expression up to permuting the first element to the end. As this requires d swaps,it changes the sign by ( − d . (cid:3) We are now ready to prove the general case of Lemma 4.9: Lemma 6.6. Let d = 2 n or n − . In the spectral sequence of (4.8) the map ǫ n − : H d ( I ( X ); Z [ ] σ ) H n − P ∗ ( X ) is induced by the map taking a chain ( g , . . . , g d ) to the sum (cid:18) d Y i =1 det( g i ) (cid:19) X σ ∈ Aut { ,...,d } sgn( σ ) [ V σ, ⊆ V σ, ⊆ · · · ⊆ V σ,d ] . Here we define V σ,i = span( h σ (0) x , h σ (1) x , . . . , h σ ( i ) x ) , with h d = 1 , h i = g d · · · g d − i , and x any fixed point in X .When k = R this is the class of the d -simplex with vertices { h x , h x , . . . , h d x } . Proof. The map ǫ n − is induced by the edge homomorphism cofib th Y hI ( X ) cofib th b Y hI ( X ) , where b Y ( ~A ) def = ( Y (( d )) if ~A = ( d ) ∗ otherwise . (This is the quotient of the n − n − I ( X )-equivariantly via the methods above. As all maps in b Y map to the basepoint, each direction will simplysuspend the ( d )-case. This implies that ǫ n − is the map on H d +1 induced by taking I ( X )-invariants of themap S σ ∧ Σ d S S σ ∧ ι S σ ∧ Σ d F X • where ι is the inclusion of the 0-skeleton into F X • .This map can be described in an alternate manner. Model homotopy coinvariants via taking an extrasimplicial direction (see, for example, [GJ99, Example IV.1.1]) and take the double chain complex associatedto a bisimplicial set. Then the homology of the geometric realization is isomorphic to the homology ofthis total complex. Due to the suspension coordinates, the bottom d + 1 rows of this double complex are0. Above this (assuming that the group-coordinate is vertical and the flag-coordinate is vertical) the mapabove includes the standard bar construction for H ∗ ( I ( X ); Z [ ] σ ) as the leftmost column. In order to showthat the given formula for ǫ n − holds it is therefore sufficient to show the following: given a cycle σ in C d ( I ( X ); Z [ ] σ ) (which lies in coordinate (0 , d + 1) in the double complex), it is homologous to the cyclegiven by the formula in the statement of the lemma (which lies in coordinate ( d, d + 1)).In the double complex associated to the above bisimplicial construction, the group at coordinate ( m, ℓ + d + 1) is generated by diagrams of the form(6.7) X X · · · X m g X g X · · · g X m ... ... ... g ℓ X g ℓ X · · · g ℓ X m We denote such a diagram by ( g , . . . , g ℓ )[ X ⊆ · · · ⊆ X m ] . We define { x | · · · | x i } = X σ ∈ Aut { ,...,i } sgn( σ )[span( x σ (0) ) ⊆ span( x σ (0) , x σ (1) ) ⊆ · · · ⊆ span( x σ (0) , . . . , x σ ( i ) ) ⊆ X ] . This double complex contains as a (vertically shifted by d + 1) subcomplex the complex C ∗∗ of Lemma 6.5;the conclusion of the lemma is exactly the desired formula.The last claim in the lemma follows by Theorem 1.20. (cid:3) ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 29 Appendix A. Comparing RT-buildings to the classical constructions In this appendix we prove our claims in Theorem 1.20 and Theorem 2.15 that the construction of thescissors congruence groups in our account agrees with the classical constructions. To begin we introducean object closely related to the configuration space of points in X . The simplices in this space are tuplesof points in X ; unlike in the configuration space, points are allowed to be repeated, and this can producenondegenerate simplices. For example, for any two distinct points a, b ∈ X , ( a, b, a ) is a nondegenerate2-simplex in Tuple m • ( X ). Definition A.1. Tuple m • ( X ) is the simplicial set whose i -simplices are given by the subset of Q ij =0 X ofthose tuples ( x , . . . , x i ) such that any subset of the tuple has a nondegenerate span of dimension at most m . The j -th face map is given by dropping the j -th element of the tuple; the j -th degeneracy is given byrepeating the j -th element of the tuple.The homology of Tuple m • ( X ) is directly related to scissors congruence groups, as the following resultsillustrate: Theorem A.2. [Dup01, Theorem 2.10] Let k = R . The map taking a tuple of points to its convex hulldefines a I ( H n ) -equivariant isomorphism H n (Tuple n • ( H n ) / Tuple n − • ( H n )) σ P ( H n , . Here, · σ means that the action is twisted by the determinant: for any g ∈ I ( H n ) , g acts on a homology onthe left by ( − det g as well as by the usual action on H n . The spherical case is more complicated. Recall the map Σ, defined as the suspension of a polytope, fromDefinition 1.13. Theorem A.3 ([Dup82, Corollary 5.18]) . The map taking a simplex to its convex hull induces a O ( n + 1) -equivariant isomorphism H n (Tuple n • ( S n ) / Tuple n − • ( S n )) σ (coker Σ) . In particular, since Σ is O ( n + 1) -equivariant, Σ induces an isomorphism on coinvariants H ( O ( n + 1) , H n (Tuple n • ( S n ) / Tuple n − • ( S n ) σ ) P ( S n , O ( n + 1)) . It follows that in order to relate scissors congruence groups and the homology of S σ ∧ F X • it suffices toshow that Tuple dim X • ( X ) / Tuple dim X − • ( X ) and F X • are I ( X )-equivariantly homotopy equivalent.We begin with a basic lemma about the homotopy type of Tuple • ( X ) in the absence of dimension restric-tions: Lemma A.4. Tuple dim X • ( X ) ≃ ∗ . Proof. By [Cat04, Proposition 2.2.1], since k is infinite e H ∗ (Tuple dim X • ( X )) = 0. (In fact, Cathelineau provesthis only with rational coefficients, but his proof works equally well integrally.) To see that Tuple dim X • ( X )is contractible it suffices to check that it is simply-connected. By [Cat04, Proposition 2.2.2] for any pair ofpoints ( x, y ) spanning a subspace of X , the subset W x,y of those points in X such that ( x, y, w ) spans asubspace is a Zariski-open subspace of X . Suppose that we are given a loop represented by the sequence of1-simplices ( x , x ) , ( x , x ) , . . . , ( x i , x ). Then, since k is infinite, there exists a point w such that ( x j , w )spans a subspace for all j , and the loop is homotopic to a loop of the form ( x , w ) , ( w, x ). This is contractedby the 2-simplex ( x , w, x ), so Tuple dim X • ( X ) is contractible. (cid:3) For any simplicial set K • , let Sd K • be the barycentric subdivision of K • [GJ99, Section III.4]. Define themap h : Sd Tuple m • ( X ) T m • ( X ) to be the map induced by taking a tuple of points in X to their span.More explicitly, an i -simplex in Sd Tuple m • ( X ) is a sequence ~x ⊆ ~x ⊆ · · · ⊆ ~x i , where ~x j is a tuple in X and ~x j − is an (ordered) subset of ~x j for all j . Taking the spans of each tuple produces an i -simplex in T m • ( X ); as taking spans is G -equivariant, this map is G -equivariant. Proposition A.5. The map h : Sd Tuple m • ( X ) T m • ( X ) induced by taking tuples in X to their spans is a G -equivariant weak equivalence. Proof. We use Theorem A’ [GG87, p.578], which states that a map of simplicial sets is a weak equivalenceif the “naive left homotopy fiber” above every simplex in the codomain is contractible. Here, for a given a q -simplex y ∈ T mq ( X ) represented by ( U ⊆ · · · ⊆ U q ), the naive left homotopy fiber is the simplicial set( h | y ) p = n ( ~x ⊆ · · · ⊆ ~x p ) ∈ Sd Tuple mp ( X ) (cid:12)(cid:12)(cid:12) all entries of ~x p are in U o . (For a precise definition of the naive homotopy fiber see for example [JKM + 04, Defn. 3.1].) In this case,( h | y ) p is isomorphic to the simplicial set Sd Tuple m • ( U ); as this is isomorphic to Sd Tuple dim U • ( U ) it iscontractible by Lemma A.4. (cid:3) The m = dim X case of the following theorem shows that F X • is G -equivariantly homotopy equivalent toa quotient of tuple spaces. Theorem A.6. For all m ≥ m • ( X ) / Tuple m − • ( X ) ≃ T m • ( X ) /T m − • ( X ) via a zigzag of G -equivariant maps.Proof. We have the G -equivariant commutative diagramTuple m − • ( X ) Sd Tuple m − • ( X ) T m − • ( X )Tuple m • ( X ) Sd Tuple m • ( X ) T m • ( X ) h ∼ h ∼∼∼ where the vertical maps are injective on all i -simplices, hence cofibrations. Taking vertical cofibers gives thedesired result, as the cofibers of the vertical maps are also the homotopy cofibers. (cid:3) Corollary A.7. e H i ( F X • ) = 0 for i = dim X. Proof. By definition F X • = T dim X • ( X ) /T dim X − • ( X ). As all simplices of T dim X • ( X ) above dim X are degen-erate (since they must repeat at least one subspace) it must be the case that e H i ( F X • ) = 0 for i > dim X . ByTheorem A.6, F X • ≃ Tuple dim X • ( X ) / Tuple dim X − • ( X ). However, all simplices of Tuple dim X • ( X ) of dimen-sion less that dim X are contained in Tuple dim X − • ( X ), since the span of i points has dimension at most i − 1. Thus e H i ( F X • ) = 0 for i < dim X . (cid:3) Using these results we can finally prove Theorem 1.20: Proof of Theorem 1.20. Theorems A.2 and A.3, together with (1.16) demonstrate that scissors congruencegroups are group homology with coefficients in H n ( F X • ) σ ; this is exactly H n +1 ( S σ ∧ F X • ). By the homotopyorbit spectral sequence (Proposition B.7) this is H n +1 (( S σ ∧ F X • ) hI ( X ) ), as desired. The formula for theclass represented by the vertices of a simplex follows from Theorem A.6 and the fact that on homology theinverse to the map Sd Tuple m − • ( X ) ∼ Tuple m − • ( X ) is given by the formula[ x , . . . , x n ] X σ ∈ Aut { ,...,n } sgn( σ ) (cid:2) ( x σ (0) ) ′ ⊆ ( x σ (0) , x σ (1) ) ′ ⊆ · · · ⊆ ( x σ (0) , . . . , x σ ( n ) ) ′ (cid:3) . Here, ( x σ (0) , . . . , x σ ( i ) ) ′ is ordered not by the ordering 0 , . . . , i but rather by the ordering induced on the x ’sfrom the tuple ( x , . . . , x n ). (cid:3) We wrap up this section by proving our claim in Theorem 2.15, that the derived definition of the Dehninvariant is compatible with the classical Dehn invariant. As we saw in Theorem 1.20, in order to translatebetween classical scissors congruence groups and RT-buildings we must take “semi-coinvariants,” and thusthe twist by S σ (Definition 1.17) appears here as well. ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 31 Proof of Theorem 2.15. Rewriting b D i using Lemma 2.4, we see that it suffices to construct an I ( X )-equivariantdiagram relating L U b D U to H n +1 ( S σ ∧ D i ).For a geometry W of dimension i , write R • ( W ) def = Sd Tuple i • ( W ) / Sd Tuple i − • ( W )for the quotient of barycentric subdivisions Sd. To define D Ri : R • ( X ) W U R • ( U ) e ⋆ R • ( U ⊥ ) consider a j -simplex of R • ( W ): this is represented by a sequence T ⊆ · · · ⊆ T j of tuples of points in W such that thespan of T j is W . If there exists a maximal ℓ such that dim span T ℓ = i , we map this j -simplex to the simplex( T ⊆ · · · ⊆ T ℓ ) ∧ (pr U ⊥ T ℓ +1 ⊆ · · · ⊆ pr U ⊥ T j ), indexed by span T ℓ . Otherwise, we map to the basepoint.This is a well-defined simplicial map for the same reason that D i is.Consider the following diagram: H n +1 ( S σ ∧ F X • ) H n +1 (cid:18) S σ ∧ _ U ⊆ X dim U = i F U • e ⋆ F U ⊥ • (cid:19) H n +1 ( S σ ∧ R • ( X )) H n +1 (cid:18) S σ ∧ _ U ⊆ X dim U = i R • ( U ) e ⋆ R • ( U ⊥ ) (cid:19) P ( X, M U ⊆ X dim U = i P ( U, ⊗ P ( S n − i − , H n +1 ( S σ ∧ D i ) h ∼ h ∼ H n +1 ( S σ ∧ D Ri ) D Q p ∼ p ∼ Here, the vertical maps h are induced by the map h in Theorem A.6 (and are thus isomorphisms). Thevertical maps p are defined as in Theorem 1.20 (with G trivial) and are therefore isomorphisms. Since allmaps in this diagram are I ( X )-equivariant, the lemma follows. (cid:3) Appendix B. Technical miscellany B.1. Reduced joins. In this section we restate the definition of a reduced join and prove several importantproperties. Definition B.1. We define X e ⋆ Y to be the simplicial set with( X e ⋆ Y ) n = n − _ i =0 X i ∧ Y n − i − . On a simplex ( x, y ) ∈ X i ∧ Y n − i − , the map d j is defined to be d j ∧ X i ∧ Y n − i − X i − ∧ Y n − i − if j ≤ i and 1 ∧ d j − i − : X i ∧ Y n − i − X i ∧ Y n − i − if j ≥ i + 1. The degeneracies are defined similarly. Lemma B.2. Reduced joins distribute over wedge products.Proof. We have _ α ∈ A ( X α e ⋆ Y ) ! n = _ α ∈ A _ i + j = n − ( X α ) i ∧ Y j = _ i + j = n − _ α ∈ A X α ! i ∧ Y j = _ α ∈ A X α ! e ⋆ Y ! n . Since each step of this expression commutes with simplicial maps, the two are isomorphic as simplicialsets. (cid:3) Lemma B.3. Let f : X Y be a quotient of simplicial sets. Then the map f e ⋆ X e ⋆ Z Y e ⋆ Z is alsoa quotient of simplicial sets. ( f e ⋆ − ( ∗ ) = f − ( ∗ ) e ⋆ Z . Proof. It suffices to show that every nonbasepoint simplex in the codomain has a unique preimage in thedomain. Consider a non-basepoint n -simplex in Y e ⋆ Z ; this is a pair of the form ( y i , z j ) with y i ∈ Y i , z j ∈ Z j and i + j = n − 1. As y i ∈ Y i is non-basepoint, it has a unique preimage x i ∈ X i . As the given map takes( x, z ) to ( f ( x ) , z ) the preimage of ( y i , z j ) is exactly ( f − ( y i ) , z j ), which is unique.The simplices that map to the basepoint are exactly those that f maps to the basepoint, with anythingin the Z -coordinate. (cid:3) We end by giving a map relating the smash product and the reduced join. Lemma B.4. Let X and Y be pointed simplicial sets. The map f : S ∧ X ∧ Y X e ⋆ Y given by sending ( i, x, y ) ∈ ( S ∧ X ∧ Y ) n to ( d n − i +1 i x, d i +10 y ) is a simplicial weak equivalence.Proof. The fact that f is well-defined is direct from the definition. We define X ∗ w Y to be the doublemapping cylinder of the diagram X pr X X × Y pr Y Y. We can thus think of X ∗ w Y as the quotient of I × X × Y given by the mapping cylinder relations ( x, , y ) ∼ ( x ′ , , y ) and ( x, , y ) ∼ ( x, , y ′ ) for all x, x ′ ∈ X and y, y ′ ∈ Y . Consider the following commutative square: X ∗ w Y S ∧ X ∧ YX ∗ Y X e ⋆ Y g fg ′ f ′ The maps g and g ′ are both weak equivalences because they are quotients by contractible subspaces. The map f ′ is a weak equivalence by [FG04, Corollary 3.4]. Thus, by 2-of-3, f is a weak equivalence, as desired. (cid:3) B.2. Homotopy coinvariants. All of the results in this section are well-known to experts, although wecould not find references for them for the specific cases we were interested in. Definition B.5 ([GJ99, Example IV.1.10]) . Let X be a (pointed) simplicial set with an action by a discretegroup G . The homotopy coinvariants (or homotopy orbits ) of G acting on X , denoted X hG , is the diagonalof the bisimplicial set with ( m, n )-simplices given by diagrams x g g x g g g x · · · g n g n · · · g x for x ∈ X m .Directly from the definition we see that ∗ hG ∼ = ∗ and S hG ∼ = BG + . Remark B.6 . This agrees with the more standard definition of homotopy coinvariants, defined as X hG def = EG + ∧ G X. (In the unpointed context, ∧ is replaced by × .)There is a spectral sequence for computing the homology of the homotopy orbits from the group homologyof G with coefficients in the homology of X : Proposition B.7. There is a spectral sequence H p ( G, e H q ( X )) e H p + q ( X hG ) . The proposition holds for all simplicial sets with G -action, which is the case of concern in this paper. Proof. Consider X as an unpointed simplicial set; write this space X . The homology of the diagonal simplicialset of a bisimplicial set is the homology of the total complex of the associated simplicial abelian group. Thespectral sequence associated to a simplicial abelian group A •• has E p,q = H vert q H horiz p ( A •• ) H p + q (diag A •• ) . Applying this in the current case to both X hG and ∗ hG gives us the following pair of spectral sequences: H p ( G, H q ( X )) H p + q ( X hG ) and H p ( G, H q ( ∗ )) H p + q ( BG ) . ILBERT’S THIRD PROBLEM AND A CONJECTURE OF GONCHAROV 33 The second is a retract of the first; if we take the other summand, we get a spectral sequence H p ( G, e H q ( X )) e H p + q ( X hG ) , as desired. (cid:3) Lastly we present a technical proposition relating certain kinds of homotopy orbits. Proposition B.8. Let G be a group acting on a pointed simplicial set X • . Suppose that Y • is a subspace of X • such that the following two conditions hold: (1) If g ∈ G is such that there exists a (non-basepoint) simplex y ∈ Y • such that g · y ∈ Y • then for all y ′ ∈ Y • , g · y ′ ∈ Y • . (2) For all n and for all x ∈ X n there exists g ∈ G such that g · x ∈ Y n .Let H be the subgroup of G that takes Y • to Y • . Then ( X • ) hG ≃ ( Y • ) hH . Proof. Let Z •• be the bisimplicial set whose ( n, m )-simplex consist of diagrams x g x g · · · g m x m , where the x i ∈ X n for i = 0 , . . . , m and g i · x i − = x i . Then diag Z •• = ( X • ) hG . In addition, if we let W •• bethe sub-bisimplicial set containing those diagrams where the x i ∈ Y • and the g i ∈ H then diag W •• = Y hH .Thus it suffices to check that the inclusion W •• Z •• induces an equivalence on diagonals. To prove this, itsuffices (by [GJ99, Proposition IV.1.9]) to show that for all n , W n • Z n • is a weak equivalence of simplicialsets. Z n • (resp. W n • ) is the nerve of the category whose objects are X n (resp. Y n ) and whose morphisms areinduced by the action of G (resp. H ); call these categories C and D . D is clearly a subcategory of C ; thusto show that the map induces an equivalence on nerves it suffices to check that the inclusion is full andessentially surjective. That it is full follows from condition (1), since since if we are given y, y ′ ∈ Y n thenany g such that g · y = y ′ is in H . That it is essentially surjective follows from condition (2), since everyelement of X n is isomorphic via the action of G to an element of Y n . (cid:3) B.3. The spectral sequence for the total homotopy cofiber of a cube. .The technical result that we need in order to understand the Dehn cube is the spectral sequence for thetotal homotopy cofiber of a cube. As the usual spectral sequence is stated only for ordinary, rather thanreduced, homology, we state our analog here. We use the notation introduced in Section 3. Proposition B.9. Let F : b I n Top ∗ be a functor. There is a spectral sequence M ~A =( b,a ,...,a n − p − ) e H q ( F ( ~A )) e H p + q (cofib th F ) . Proof. By [MV15, Proposition 9.6.14], for a functor G : b I n Top there is a spectral sequence M ~A =( b,a ,...,a n − p − ) H q ( G ( ~A )) H p + q (cofib th G ) . Each of the spaces we have is pointed, thus the functor C : b I n Top defined by C ( ~A ) = ∗ is a retract of G . In particular, this means that the spectral sequence given by the kernel of the induced map G C isalso a spectral sequence, which converges to ker( H p + q (cofib th G ) H p + q (cofib th C )). Since cofib th C ≃ ∗ ,this reduces to the desired spectral sequence. (cid:3) References [BG02] Jos´e I. Burgos Gil, The regulators of Beilinson and Borel , CRM Monograph Series, vol. 15, American MathematicalSociety, Providence, RI, 2002. MR 1869655[BKSOsr15] A. J. Berrick, M. Karoubi, M. Schlichting, and P. A. Ø stvær, The Homotopy Fixed Point Theorem and theQuillen-Lichtenbaum conjecture in Hermitian K -theory , Adv. Math. (2015), 34–55. MR 3341783[Bro82] K.S. Brown, Cohomology of groups , Graduate Texts in Mathematics, Springer, 1982.[Cat03] Jean-Louis Cathelineau, Scissors congruences and the bar and cobar constructions , J. Pure Appl. Algebra (2003), no. 2-3, 141–179. MR 1975297 [Cat04] , Projective configurations, homology of orthogonal groups, and milnor k -theory , Duke Math. J. (2004),no. 2, 343–387.[Cat07] , Homology stability for orthogonal groups over algebraically closed fields , Annales scientifiques de l’´EcoleNormale Sup´erieure Ser. 4, 40 (2007), no. 3, 487–517 (en).[CS85] Jeff Cheeger and James Simons, Differential characters and geometric invariants , Geometry and topology (CollegePark, Md., 1983/84), Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 50–80. MR 827262[CZ] Jonathan Campbell and Inna Zakharevich, D´evissage and localization for the grothendieck spectrum of varieties ,arXiv: 1811.08014.[DHZ00] Johan Dupont, Richard Hain, and Steven Zucker, Regulators and characteristic classes of flat bundles , The arith-metic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc.,Providence, RI, 2000, pp. 47–92. MR 1736876[DK90] Johan L. Dupont and Franz W. Kamber, On a generalization of Cheeger-Chern-Simons classes , Illinois J. Math. (1990), no. 2, 221–255. MR 1046564[DS82] Johan L. Dupont and Chih Han Sah, Scissors congruences. II , J. Pure Appl. Algebra (1982), no. 2, 159–195.MR 662760[Dup82] Johan L. Dupont, Algebra of polytopes and homology of flag complexes , Osaka J. Math. (1982), no. 3, 599–641.[Dup01] , Scissors congruences, group homology and characteristic classes , Nankai Tracts in Mathematics, vol. 1,World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1832859[FG04] R Fritsch and Marek Golasinski, Topological, simplicial and categorical joins , Archiv der Mathematik (2004),468–480.[GG87] Henri Gillet and Daniel R. Grayson, The loop space of the Q -construction , Illinois J. Math. (1987), no. 4,574–597. MR 909784[GJ99] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory , Progress in Mathematics, vol. 174, Birkh¨auserVerlag, Basel, 1999. MR 1711612 (2001d:55012)[Gon99] Alexander Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives , J. Amer. Math. Soc. (1999),no. 2, 569–618. MR 1649192[GR18] Alexander B. Goncharov and Daniil Rudenko, Motivic correlators, cluster varieties and zagier’s conjecture onzeta(f,4) , 2018.[Gra94] Daniel R. Grayson, Weight filtrations in algebraic K -theory , Motives (Seattle, WA, 1991), Proc. Sympos. PureMath., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 207–237. MR 1265531[Jes68] Børge Jessen, The algebra of polyhedra and the Dehn-Sydler theorem , Math. Scand. (1968), 241–256 (1969).MR 0251633 (40 + 04] Kevin Charles Jones, Youngsoo Kim, Andrea H. Mhoon, Rekha Santhanam, Barry J. Walker, and Daniel R.Grayson, The additivity theorem in K -theory , K -Theory (2004), no. 2, 181–191. MR 2083580[Kar80] Max Karoubi, Le theoreme fondamental de la k-theorie hermitienne , Annals of Mathematics (1980), no. 2,259–282.[Knu12] K.P. Knudson, Homology of linear groups , Progress in Mathematics, Birkh¨auser Basel, 2012.[MV15] Brian A. Munson and Ismar Voli´c, Cubical homotopy theory , New Mathematical Monographs, vol. 25, CambridgeUniversity Press, Cambridge, 2015. MR 3559153[Qui73] Daniel Quillen, Finite generation of the groups K i of rings of algebraic integers , Algebraic K -theory, I: Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 1973, pp. 179–198. Lecture Notes in Math.,Vol. 341. MR 0349812[Rie] Emily Riehl, Homotopical categories: From model categories to ( ∞ , -categories , arXiv:1904.00886.[Sah79] C. H. Sah, Hilbert’s third problem: scissors congruence , Research Notes in Mathematics, vol. 33, Pitman (AdvancedPublishing Program), Boston, Mass., 1979. MR 554756 (81g:51011)[Syd65] J.-P. Sydler, Conditions n´ecessaires et suffisantes pour l’´equivalence des poly`edres de l’espace euclidien `a troisdimensions , Comment. Math. Helv. (1965), 43–80. MR 0192407 (33 The K -book , Graduate Studies in Mathematics, vol. 145, American Mathematical Society,Providence, RI, 2013, An introduction to algebraic K -theory. MR 3076731[Zak17] Inna Zakharevich, The K -theory of assemblers , Advances in Mathematics304