aa r X i v : . [ m a t h . G T ] D ec THE SPACE OF CLOSED SUBGROUPS OF R n by Benoˆıt Kloeckner
Abstract . —
The Chabauty space of a topological group is the set of its closedsubgroups, endowed with a natural topology. As soon as n >
2, the Chabauty spaceof R n has a rather intricate topology and is not a manifold. By an investigation ofits local structure, we fit it into a wider, but not too wild, class of topological spaces(namely Goresky-MacPherson stratified spaces). Thanks to a localization theorem,this local study also leads to the main result of this article: the Chabauty space of R n is simply connected for all n . Last, we give an alternative proof of the Hubbard-Pourezza Theorem, which describes the Chabauty space of R .
1. Introduction
Let G be a topological group whose neutral element is denoted by 0 (although G need not be abelian). Its Chabauty space C ( G ) is the set of closed subgroups of G endowed with the following topology: the neighborhoods of a point Γ ∈ C ( G ) are thesets N KU (Γ) = { Γ ′ ∈ G | Γ ′ ∩ K ⊂ Γ · U and Γ ∩ K ⊂ Γ ′ · U } where K runs over the compact subsets of G and U runs over the neighborhoods of0. In words, Γ ′ is very close to Γ if, on a large compact set, every of its elements isin a uniformly small neighborhood of an element of Γ, and conversely. The preprint[ ] contains a more detailed account of this topology.The Chabauty space is named after Claude Chabauty, who introduced it in [ ] togeneralize Mahler’s compactness criterion to lattices in locally compact groups. If G is locally compact, then C ( G ) is compact and can therefore be used to define a com-pactification of any space whose points are naturally associated to closed subgroupsof G . For example, this is the case of a symmetric space of noncompact type: onemaps a point to its stabilizer in the isometry group. The corresponding compactifica-tion is isomorphic to the Satake compactification [
16, 1 ]. This compactification wasgeneralized to buildings thanks to the Chabauty topology point of vue in [ ].The simplest example of a Chabauty space is that of the line: C ( R ) contains thetrivial subgroup { } , the discrete groups α Z and the total group R . Two discrete BENOˆIT KLOECKNER groups α Z and β Z are close one to another when α and β are close, a neighborhoodof { } consists in the set of α Z with large α (and we define ∞ Z = { } ) and aneighborhood of R consists in the set of α Z with small α (and we define 0 Z = R ).Putting all this together, we see that C ( R ) is homeomorphic to a closed interval. α Z { } = ∞ ZR = 0 Z Figure 1.
Chabauty space of R . Only for a few groups G is known a precise description of C ( G ). Recent work ofBridson, de la Harpe and Kleptsyn adds to the list the three-dimensional Heisenberggroup [ ], but the topology of C ( R n ) is terra incognita for n >
2. Even C ( R ) isuneasy to describe; it was tackled by Hubbard and Pourezza [ ] who proved thefollowing. Theorem A (Hubbard-Pourezza) . —
Let C be the Chabauty space of R and L be the subset of lattices. The topological pair ( C , C r L ) is homeomorphic to thesuspension of ( S , K ) where K is a trefoil knot in the -sphere. In particular, C is a -sphere. Let us recall some definitions. A topological pair is a pair (
X, Y ) of topologicalspaces where Y is a subset of X (endowed with the induced topology). Two topologicalpairs ( X, Y ) and ( X ′ , Y ′ ) are homeomorphic if there is a homeomorphism Φ : X → X ′ that maps Y onto Y ′ . The (open) cone over X is the quotient cX of X × [0 ,
1) by therelation ( x , ∼ ( x , suspension of X is the quotient sX of X × [0 ,
1] bythe relations ( x , ∼ ( x ,
0) and ( x , ∼ ( x ,
1) for all x , x ∈ X . If Y is a subsetof X , then sY embeds naturally in sX and the resulting topological pair ( sX, sY ) iscalled the suspension of ( X, Y ).The Hubbard-Pourezza theorem shows in particular that the set of non-latticesis a 2-sphere that is non-tamely embedded in C ( R ). At the end of the paper weshall give a proof of Theorem A using Seifert fibrations. It is likely to be a variationof the alternative proof alluded to in [ ], but we provide it so that this article isself-contained; we shall indeed see that Theorem A gives information on the “links”of some points in C ( R n ) for higher n .Our main goal is to investigate the space C ( R n ). It is not a manifold when n > Theorem B . —
For all n , the Chabauty space of R n admits a Goresky-MacPhersonstratification. If n > it is moreover a pseudo-manifold. We shall give Goresky and MacPherson’s definitions of a stratification and pseudo-manifold later on; roughly, it means that C ( R n ) is a union of manifolds nicely gluedtogether. Compact Goresky-MacPherson stratified spaces have for example a well-defined intersection homology, locally contractible homeomorphism group and exten-sion of isotopy properties. HE SPACE OF CLOSED SUBGROUPS OF R n To prove Theorem B we shall unveil part of the local topology of C ( R n ). Its globaltopology seems difficult to make completely explicit. It might be possible to describeprecisely C ( R ), but as n grows the space becomes more and more complicated. Evenif one describes each of its strata, the way they glue together is quite involved (evenif locally trivial in some sense). In such a case, one tries to compute some topologicalinvariants to get a grip on the space, the first one being its fundamental group. Ourmain result is the following. Theorem C . —
For all n , the Chabauty space of R n is simply connected. Let us give a sketch of the proof of this result; in the sequel we often use the notation C := C ( R n ). The subset R m ⊂ C of maximal rank subgroups (that is, subgroupscontaining a basis of vectors of R n ) is open, dense and contractible. Its complement R ℓ := C r R m is a subspace of codimension n . If C where a differentiable manifold,we could have proceeded by tansversality arguments: any loop based in a point of R m would be homotopic to a generic smooth loop, transversal to R ℓ . But transversalitybetween a submanifold of codimension > R m is contractible the loop would then be nullhomotopic.One must be very carefull when trying to apply these arguments in more generalspaces. For example, the cone over a disconnected manifold is a stratified space, andit is not true that a generic curve avoids its apex. Figure 2.
A generic curve does not avoid the apex.
This example is however very local in nature, and one guesses that if no such phe-nomenon occurs, then one should be able to proceed almost as if C where a manifold.This guess is true, as is shown by the following relative homotopy localization result. Theorem D (localization) . —
Let X be a Hausdorff topological space and Y be aclosed subset of X .If every point y ∈ Y admits a neighborhood system ( U ε ) ε such that each pair ( U ε , U ε r Y ) is k -connected, then the pair ( X, X r Y ) is k -connected. In other words, under very mild assumptions a pair that is locally k -connected mustbe globally k -connected. Let us recall that a pair ( X, U ) is 0-connected if any pointin X can be connected by a continuous path to a point in U . The pair is k -connectedif moreover for all ℓ k , every map ( I ℓ , ∂I ℓ ) → ( X, U ) from the closed cube I ℓ thatmaps its boundary into U is homotopic (relative to its boundary) to a map I ℓ → U .It would be very surprising that such a simple and helpful result be new, howeverI could not find a reference in the litterature (except [ ] where it is proved only forpolyhedral pairs) and we shall therefore provide a proof. BENOˆIT KLOECKNER
The topics of the next sections are: some preliminaries and definitions (Section 2),stratifications (Section 3), the local study of C ( R n ) –including the proof of TheoremB (Section 4), proofs of Theorems C and D (Section 5), alternative proof of TheoremA (Section 6), and some open questions (Section 7). Aknowledgements . — I wish to thank Lucien Guillou for topological discussions,Patrick Massot for patiently explaining Seifert fibration to me, and Pierre de la Harpefor both his comments on this article and his talk during the fourteenth Tripodemeeting in Lyon, which made me discover the nice topic of Chabauty spaces.
2. Types and norms
We fix n ∈ N and consider the Chabauty space C = C ( R n ).Let · denote the canonical scalar product on R n and | | denote the correspondingEuclidean norm. It defines a distance not only on R n , but also on the Grassmannian G ( p ; n ) of all its p -dimensional sub-vector spaces ( p -planes, for short). There areseveral classical ways to do this, but they do not differ for our purpose.We denote under brackets h i the vector space generated by a subset of R n . Let Γ be a point in C . It is isomorphic to R p × Z q for some integers p, q . The pair ( p, q ) is called the type of Γ. The rank of Γ is the dimension of thevector space h Γ i it generates, that is p + q . An element of rank < n is said to beof lower rank. We denote by R ℓ the set of lower rank elements of C , by R m itscomplement and by C ( p,q ) the set of type ( p, q ) elements. The Lie group GL( n ; R )acts naturally on C and its orbits are exactly the C ( p,q ) . Let Γ be a type ( p, q ) point in C . For all positive r , let Γ( r ) be thesubgroup of R n generated by Γ ∩ B (0 , r ) where B (0 , r ) is the closed ball of radius r centered at the origin. Let Γ := ∩ r> Γ r be the continuous part of Γ. It is a p -planeof R n .If p >
0, then define N (Γ) = . . . = N p (Γ) = 0 . Let r be the least number r such that Γ( r ) = Γ and p be the rank of Γ( r ). Thendefine N p +1 (Γ) = . . . = N p (Γ) = r . Define similarly ( r , p ) , . . . , ( r k , p k ) until Γ( r k ) = Γ. Then one has q = p k − p . Atlast, define N p + q +1 (Γ) = . . . = N n (Γ) = ∞ . The number N i (Γ) is called the i -th norm of Γ. The norms are continuous functions N i : C → [0 , ∞ ]. Any Γ ∈ C has a canonical decomposition Γ = Γ + Γ D ,where Γ D := Γ ∩ Γ ⊥ is a discrete rank q subgroup of R n . HE SPACE OF CLOSED SUBGROUPS OF R n Γ D Figure 3.
Canonical decomposition of a type (1 ,
1) subgroup of R . Let us consider the incidence scheme of thedifferent C ( p,q ) ; these subsets are natural “strata” of C .If n = 1, there are three types, namely those of R , Z and the trivial group, fromnow on denoted by 0. The closure of the second one contains the two other (bothreduced to a point). We sum this up into the diagram: Z (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ????? R n = 2, there are six types, organized according to the diagram: Z (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ????? R × Z (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ????? Z (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ????? R R Z n (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ????? R × Z n − (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ????? Z n − (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ????? · · · (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) · · · · · · (cid:31) (cid:31) ????? R n − × Z (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127) (cid:31) (cid:31) ???? (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) · · · (cid:31) (cid:31) ?????? Z (cid:31) (cid:31) ????? (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) R n R n − · · · R BENOˆIT KLOECKNER to be read as follows: the closure of the orbit of type ( r, s ) intersects the orbit oftype ( p, q ) if and only if there is a sequence of arrows R r × Z s → · · · → R p × Z q (inwhich case we write ( r, s ) > ( p, q )). A sequence of type ( r, s ) elements can indeedconverge to a point of a different type in two (possibly simultaneous) ways: some ofthe non-zero, finite N i go to 0 or to ∞ . In both cases s decreases; in the first one therank is constant and r increases while in the second one r is constant and the rankdecreases. In other words, ( r, s ) > ( p, q ) if and only if r p and r + s > p + q : this isexactly what the diagram tells.Note that for example each (lower left)-(upper right) diagonal corresponds to thesubgroups of a given rank. In particular the largest of these diagonals corresponds tothe set R of higher rank subgroups. Similarly, each (upper left)-(lower right) diagonalcorresponds to the subgroups with continuous part of a given dimension, in particularthe largest of these diagonals correspond to the set of discrete subgroups. There is a well-known duality on the space of lattices of R n . Itextends word by word to the larger space C ( R n ): the duality map ∗ : C ( R n ) → C ( R n )Γ Γ ∗ = { y ∈ R n | ∀ x ∈ Γ x · y ∈ Z } is an involutory homeomorphism. The dual of a type ( p, q ) element is of type ( n − ( p + q ) , q ). In particular, in the terminology to be introduced in the next section, ∗ is a stratified isomorphism. On the above diagrams, duality induces a reflexionwith respect to a vertical axis and one only needs to understand half of the types tounderstand them all. 0basisOrthonormal Figure 4. In R , three type (1 ,
1) subgroups (dark lines) and their duals(white points).
3. Stratifications
There are many different types of stratifications; we shall use that of Mark Goreskyand Robert MacPherson [ ], but we also introduce more general definitions and thatof Larry Siebenmann [ ]. HE SPACE OF CLOSED SUBGROUPS OF R n Definition 3.1 . — Let X be a metrizable separable topological space. A stratifica-tion of X is a locally finite partition S = ( X ( s ) ) s ∈ S into locally closed subsets called strata such that the frontier condition holds: for all s, t in S , if X ( t ) ∩ X ( s ) = ∅ then X ( t ) ⊂ X ( s ) . In other words, the closure of a stratum is a union of strata. The couple( X, S ), often simply denoted by X , is called a stratified space .In the works of Siebenmann and Goresky and MacPherson, the stratification arefiltered by { , , . . . , n } rather than a more general set S . However, in the consideredcases (CAT stratifications of finite dimension, see below) one can recover such afiltration, so that the above definition is in fact consistent with [ ] and [ ].The point in stratifying a space is to divide it into simple pieces, and the stratashould not be arbitrary for the stratification to be of interest. Most of the time, oneasks that the strata belong to a category of manifolds and think of stratified spacesas an extension of manifolds that includes some singularities. The main motivationwhen Hassler Whitney introduced the first definition of a stratified space was tostudy the topology of analytic varieties [
20, 21, 22 ]. Ren´e Thom used this conceptto investigate the smooth maps between manifolds and their singularities [ ]. Denoterespectively by TOP, PL and DIFF the categories of topological, piecewise linear andsmooth manifolds. Definition 3.2 . — Let CAT be a category of manifolds (TOP, PL, or DIFF). Astratification ( X ( s ) ) s ∈ S is a CAT stratification if all strata are objects of CAT and X ( t ) ⊂ X ( s ) implies dim X ( t ) < dim X ( s ) .The dimension d of a CAT stratification is the supremum of the dimensions of thestrata (possibly ∞ ). Its singular codimension is the difference d − d ′ where d ′ is thesecond largest dimension of the strata.The frontier condition is not very surprising since it is similar to a property ofpolyhedra (that is, topological realisation of simplicial complexes), where the closureof a face is a union of faces. To show its particular relevance, let us introduce arelation on the index set S of a stratification. Given two strata X ( s ) and X ( t ) , onewrites t s if X ( t ) ∩ X ( s ) = ∅ . It is easy to see that the frontier condition impliesthat is an ordering (the local closedness of strata is needed as well). The class of stratified spaces isstable under several natural operations. The fact that the partitions given below aregenuine stratifications is straightforward.
Definition 3.3 . — If X and Y are stratified spaces with stratifications S = ( X ( s ) ) s ∈ S and T = ( Y ( t ) ) t ∈ T the product X × Y is defined as the usual topological product endowed with thestratification S × T = (cid:16) X ( s ) × Y ( t ) (cid:17) ( s,t ) ∈ S × T BENOˆIT KLOECKNER
Every open subset U of X inherits of the induced stratification ( U ∩ X ( s ) ) s ∈ S ′ where S ′ is the set of indices s such that X ( s ) meets U .The open cone cX = X × [0 , /X × { } of a compact stratified space has a naturalstratification, whose strata are the apex and the products X ( s ) × (0 , cX iscalled a stratified cone . The cone on the empty set is defined to be a point.Recall that the join of the topological spaces X and Y is the quotient of X × Y × [0 , x, y, ∼ ( x ′ , y,
0) and ( x, y, ∼ ( x, y ′ ,
1) (figure 5 shows examples).We simply denote by A × { } the image in this quotient of a set A × Y × { } when A ⊂ X . When X and Y are stratified, their join X ⋆ Y can be endowed with anatural stratification. Let
S ⋆ T be the disjoint union of S , T and S × T ; the desiredstratification S ⋆ T is indexed by S ⋆ T , with strata(
X ⋆ Y ) ( s ) = X ( s ) × { } ( X ⋆ Y ) ( t ) = Y ( t ) × { } ( X ⋆ Y ) ( s,t ) = X ( s ) × Y ( t ) × (0 , S , a pair of distinctpoints. Figure 5.
The join of two segments is a 3-simplex. The join of two circlesis a 3-sphere, the contracted part being two Hopf fibers.
Next we need to define isomorphisms.
Definition 3.4 . — If X and Y are stratified spaces, a continuous map f : X → Y is said to be stratified if the inverse image of every stratum of Y is a stratum of X . It is a stratified isomorphism (or simply, an isomorphism) if it is stratified and ahomeomorphism.One defines in an obvious way PL and DIFF stratified maps and isomorphisms .We are now able to introduce a local triviality condition on which the notions ofSiebenmann and Goresky-MacPherson stratified spaces rely.
Definition 3.5 . — A stratified space X is locally cone-like if for any point x ∈ X ,there is an open neighborhood U of x in its stratum X ( s ) , a stratified cone cL and an HE SPACE OF CLOSED SUBGROUPS OF R n isomorphism of U × cL onto an open neighborhood of x in X such that U × { apex } ismapped identically onto U . The stratified space L is called a link of x (it need not beunique since there exist non homeomorphic spaces whose cones are homeomorphic).A Siebenmann stratified space is a finite-dimensional, locally cone-like, TOP strat-ified space.A
Goresky-MacPherson stratified space of dimension n is defined recursively as a n -dimensional Siebenmann stratified space, whose points admit links that are lesser-dimensional Goresky-MacPherson stratified spaces. One can define similarly PL andDIFF Siebenmann and Goresky-MacPherson stratified spaces.A pseudo-manifold is a Goresky-MacPherson stratified space where the union ofmaximal dimensional strata is dense, and whose singular codimension is at least 2.What we call a Siebenmann stratified space is a “CS set” in Siebenmann’s terminol-ogy. Due to the numerous definitions introduced by different authors, it seems betterto use the authors’ names to distinguish between them. Note that in [ ], contrary to[ ], the stratified spaces considered are PL.It seems to be an open question whether it exists a Siebenmann stratified spacethat is not Goresky-MacPherson stratified.Simple examples of Goresky-MacPherson stratified spaces are manifold with bound-ary (the strata being the interior and the boundary) and polyhedron (stratified bytheir faces). A consequence of Thom’s “first isotopy lemma” is that analytic varietiescan be endowed with a Goresky-MacPherson stratification, see for example [ ] SectionI.1.4. In particular, complex analytic varieties are pseudo-manifolds. Compact Siebenmann stratified spaces have several niceproperties. For example, their homeomorphism groups are locally contractible. See[ ] for more details. More important to us, it gives a very natural way to describe C = C ( R n ) locally: the types will index the strata C ( p,q ) and since their dimensionis easy to compute, the description of neighbohoods of a point reduces to a link.Among Goresky-MacPherson stratified spaces, pseudo-manifolds are of utmost im-portance since they have a so-called intersection homology satisfying some sort ofPoincar duality. It encodes in particular the usual homology. Since it is not muchmore difficult to prove that C ( R n ) is Goresky-MacPherson than to prove that it isSiebenmann, it seemed better to use this definition even if we do not compute theintersection homology of C ( R n ). Note that the codimensions of the strata need notbe even, so that there is no self-dual perversity for C ( R n ).Let us turn to a remarkable fact: the product of two Siebenmann stratified spaceis Siebenmann stratified. The only part that is not obvious in this statement is thata product of two cones is again a cone. Lemma 3.6 . —
Let X and Y be stratified spaces. Then we have an isomorphism ofstratified spaces cX × cY ≃ c ( X ⋆ Y ) .Proof . — We write the elements of cX in the form ( x, h ) where h ∈ [0 ,
1[ and x ∈ X ,the latter being meaningless when h = 0. BENOˆIT KLOECKNER
The subset { (( x, h ) , ( y, ℓ )) ∈ cX × cY ; h + ℓ < } is isomorphic to cX × cY , and isthe cone over the subset ∆ = { (( x, h ) , ( y, ℓ )) ∈ cX × cY ; h + ℓ = δ } for any δ < X ⋆ Y , simply consider the map X × Y × [0 , δ ] → ∆( x, y, m ) (( x, m ) , ( y, δ − m ))(see figure 6). X ⋆ YYX ∆ cX × cY ≃ Figure 6.
The product of cones is a cone.
Since a join is locally a product (possibly involving a cone), the join of two Sieben-mann stratified spaces is again Siebenmann stratified. Then a stratightforward induc-tion leads to : the product, the join and the cone of Goresky-MacPherson stratifiedspaces are Goresky-MacPherson stratified.
Some of the links in C ( R n ) shall be described as somesort of fiber bundles, where the fiber depends upon the strata. Note that we definesuch bundles only in the category of Goresky-MacPherson stratified spaces. Definition 3.7 . — A stratified bundle is defined inductively as a surjective contin-uous map π : E → B where: – E is a metrizable separable topological space called the total space , – B is a Goresky-MacPherson stratified space (with stratification ( B ( s ) ) s ∈ S ) calledthe base , – there are topological manifolds F s called the fibers such that all x ∈ B (in the s strata, with link L say) has a conical neighborhood V ≃ R k × cL such that π − ( V ) ≃ R k × F s × cL ′ where L ′ is a compact Goresky-MacPherson stratifiedspace and π writes in the form π ( b, f, ( t, l ′ )) = ( b, ( t, π ′ ( l ′ ))) ∀ b ∈ R k , ∀ f ∈ F s , ∀ ( t, l ′ ) ∈ cL ′ where π ′ : L ′ → L is a stratified bundle.One defines similarly PL or DIFF stratified bundles in the category of PL or DIFFGoresky-MacPherson stratified spaces. HE SPACE OF CLOSED SUBGROUPS OF R n We could have assumed E to be a Goresky-MacPherson stratified space and π tobe a stratified map, but this is not necessary. Lemma 3.8 . — If π : E → B is a stratified bundle, the partition E ( s ) = π − ( B ( s ) ) is a Goresky-MacPherson stratification.Proof . — First, let us show that the frontier condition holds. Let s and t be indicessuch that E ( t ) ∩ E ( s ) = ∅ and z ∈ E ( t ) . Then B ( t ) ⊂ B ( s ) and there is a sequence x n ∈ B ( s ) that converges to π ( z ).Thanks to the local form of π , we can lift x n to asequence z n ∈ E ( s ) that converges to z .Local closedness of strata and local finiteness of the partition are direct conse-quences of the definition, as well as the strata being topological manifolds and satis-fying the dimension condition. The partition ( E ( s ) ) s ∈ S is therefore a TOP stratifica-tion.The local form of π implies readily that it is also a Siebenmann stratification, andan induction on the dimension of B shows at last that it is Goresky-MacPherson.This definition of a stratified bundle is a generalization of fiber bundles over mani-folds, since the restriction of π to E ( s ) → B ( s ) is a fiber bundle in the usual sense forall s ∈ S . It is however quite restrictive, in particular the family of fibers cannot bearbitrary: if s > t , then F s must be homeomorphic to F t × F ′ s where F ′ s is the fiberover L ( s ) for the bundle π ′ . In the stratified bundles that appear in the local studyof C ( R n ), the fibers are tori whose dimension depends upon the strata.
4. Local study of C ( R n ) C ( R n ) . — Let us start with the simplest part of TheoremB.
Proposition 4.1 . —
The partition ( C ( p,q ) ) ( p,q ) ∈ S is a DIFF stratification of C .Proof . — First, C is known to be metrizable and compact (see for example [ ]).To see that the strata are locally closed, it is sufficient to have a look at a neigh-borhood U of a point Γ ∈ C ( p,q ) that is sufficiently small to ensure that N p +1 , . . . , N p + q ∈ (0 , ∞ )for all elements of U . There the strata is defined by the equations N = · · · = N p = 0and N p + q +1 = · · · = N n = ∞ , thus U ∩ C ( p,q ) is closed in U .The frontier condition, anyway simple to get from a direct study, comes for freefrom the description of strata as orbits of the action of GL( n ; R ): if C ( p,q ) intersects C ( r,s ) , then there is a sequence Γ n ∈ C ( r,s ) that converges to some Γ ∞ ∈ C ( p,q ) . Forall Γ ∈ C ( p,q ) there is a g ∈ GL( n ; R ) such that Γ = g (Γ ∞ ), and the sequence g (Γ n )converges to Γ, hence C ( p,q ) ⊂ C ( r,s ) .We also get the manifold structures on strata from this action: for all ( p, q ), thestabilizer of the element R p × Z q = e R + · · · + e p R + e p +1 Z + · · · + e p + q Z (where ( e i )is the canonical basis of R n ) of C ( p,q ) is a closed subgroup H ( p,q ) of GL( n ; R ), thus aLie subgroup. We can endow C ( p,q ) with the manifold structure of GL( n ; R ) /H ( p,q ) . BENOˆIT KLOECKNER
Last, the dimension of C ( p,q ) is easily computed: the continuous part is an elementof G ( p ; n ), which has dimension p ( n − p ), and the discrete part is defined by the choiceof q vectors in a ( n − p ) plane. We get thatdim C ( p,q ) = ( p + q )( n − p )in particular dim C ( r,s ) > dim C ( p,q ) as soon as ( r, s ) > ( p, q ).Note that in the sequel it will be simpler to prove only the TOP stratification oflinks, so we mainly think of C ( R n ) as a TOP stratified space. Let us introduce a number of defi-nitions to be used in the next subsection. They aim to give a parametrization ofneighborhoods in C ( R n ), by decomposing subgroups at three scales. We could definemore general definitions, involving more different scales but the following is sufficientfor our purpose. δ -decomposability . — A scale is a number δ ∈ (0 , ∈ C ( R n ) is said to be decomposable at scale δ if for all i , N i (Γ) / ∈ { δ, δ − } . We thensay that Γ has δ -type ( p, q ) if N (Γ) , . . . , N p (Γ) < δN p +1 (Γ) , . . . , N p + q (Γ) ∈ ( δ, δ − ) N p + q +1 (Γ) , . . . , N n (Γ) > δ − Note that the δ -type of Γ is always at most its type (with respect to the order ofSection 2.4, that is the order given by the frontier condition). . — The motivation for this paragraph is the following. Weshall associate to a δ -decomposable element a triple of vector spaces, generated bythree parts of Γ (one at small scale, one at medium scale and one at large scale).To compare close δ -decomposable elements, we need to fix an identification betweenclose subspaces of R n .A linear decomposition of type ( p, q ) of R n is a triple ( V , V , V ) where V is a p -plane, V is a q -plane, V is a ( n − ( p + q ))-plane and R n = V ⊥ ⊕ V ⊥ ⊕ V A linear decompositions of type ( p, q ) can be naturally identified with the ( p, p + q )flag ( V , V + V ). We therefore denote by G ( p, p + q ; n ) the set of all type ( p, q ) lineardecomposition. It is a manifold, and inherits a metric from the Euclidean structureof R n .Given a type ( p, q ) linear decomposition ( V , V , V ), there is a small ball V in G ( p, p + q ; n ) centered at ( V , V , V ) and a small ball U around the identity in asubmanifold of SO( n ) such that for all ( V , V , V ) in V , there is a unique τ ∈ U (calledthe trivialisation of ( V , V , V )) such that τ ( V , V , V ) = ( V , V , V ). Moreover themapping ( V , V , V ) τ can be chosen a diffeomorphism. From now on, we assumethat for all ( V , V , V ) we have chosen such a mapping (called a local trivialisation ).Let Γ be a δ -decomposable element of δ -type ( p, q ). The linear decomposition (atscale δ ) of Γ is defined as follows. First, V = h Γ( δ ) i is the p -plane generated by HE SPACE OF CLOSED SUBGROUPS OF R n the element of γ of norm less than δ . We denote by P ′ the orthogonal projection on V ⊥ . Then V = h P ′ Γ( δ − ) i is a q -plane orthogonal to V . At last, V is defined as( V + V ) ⊥ , and by construction ( V , V , V ) is a linear decomposition. . — Let us define a parametrization of aneighborhood of a type ( p, q ) element Γ in C ( R n ). Let V = Γ be its continuouspart and V = h Γ D i be the q -plane generated by its discrete part (which is orthogonalto V ). We define V = ( V + V ) ⊥ and we assume that a basis ( e p +1 , . . . , e p + q ) ofΓ D has been fixed. In what follows the dependence on this basis is not crucial.For convenience, we also assume that we have fixed linear isomorphisms V ≃ R p ,Γ D ≃ Z q (identifying ( e p +1 , . . . , e p + q ) with the canonical basis) and V ≃ R n − ( p + q ) .We may use this identifications without notice.Choose a small scale δ . The required smallness will be precised at several stepsbelow. First we assume that δ < N p +1 (Γ ) and δ − > N p + q (Γ ), so that Γ has δ -type ( p, q ) and ( V , V , V ) is its linear decomposition at scale δ .Then define U as the set of all Γ ∈ C ( R n ) such that: – Γ is δ -decomposable, – its linear decomposition ( V , V , V ) is δ -close to ( V , V , V ), – denoting by τ the corresponding trivialisation and by P the orthogonal pro-jection onto V , τ P (Γ( δ − )) ⊂ V is generated by vectors v p +1 , . . . , v p + q suchthat | e i − v i | < δ for all i ,It is an open neighborhood of Γ . From a Γ ∈ U we construct its local decomposition (Γ , Γ , Γ , Φ , Φ ) as follows. First, Γ = τ Γ( δ ) is a closed subgroup of V ≃ R p ofmaximal rank in C ( R p ). Second, Γ = τ P (Γ( δ − )) is a discrete subgroup of V closeto Γ D ≃ Z q . The distinguished basis of Γ is the basis v p +1 , . . . , v p + q that satisfies | e i − v i | < δ for all i (we assume δ is small enough to ensure that this basis is uniquelydefined). This distinguished basis defines an identification between Γ and Z q . Third,denoting by P the orthogonal projection onto V , Γ = τ P (Γ) is a discrete subgroupof V ≃ R n − ( p + q ) .Now Φ : Z q → V / Γ is the unique homomorphism such that Γ( δ − ) is generatedby the sets τ − ( v i + Φ ( e i )) for p < i p + q . Note that here we consider Φ ( e i ) as aΓ coset in V . Figure 7 illustrates this map. Last, Φ : Γ → ( V + V ) / (Γ + Γ )(where the range can be identified with R p / Γ × R q / Z q )) is the unique homomorphismsuch that Γ is the union of the τ − ( v + Φ ( v )) for v ∈ Γ . It can be useful to furtherdecompose Φ into Φ ′ : Γ → V / Γ and Φ ′′ : Γ → V / Γ ≃ R q / Z q .From the linear and local decompositions of Γ, it is easy to reconstruct Γ. Themap Γ (cid:0) ( V , V , V ) , (Γ , Γ , Γ , Φ , Φ ) (cid:1) and its inverse are moreover continuous.It will be simpler in the sequel to use the part of U defined by N p (Γ ) + N p + q +1 (Γ ) − < δ (note that N p + q +1 (Γ ) is the first norm of Γ viewed as an element of C ( R n − ( p + q ) )).This neighborhood is denoted by U nδ (Γ ). BENOˆIT KLOECKNER projection encodedby Φ V Γ( δ − )Γ( δ ) P Γ( δ − ) δδ − Figure 7.
The map Φ enables the recovering of Γ( δ − ) from Γ , Γ andthe local trivialisation τ (in this example, Γ has δ -type (1 , C ( R n ) . — Now are now ready to prove Theorem B.
Lemma 4.2 . —
The trivial subgroup ∈ C ( R n ) has a neighborhood of the form cL n (0) where the link L n (0) is the set of subgroups of unit (first) norm, stratified byits intersection with the strata of C ( R n ) .Proof . — The neighborhood U n (0) defined above is the set of elements of normgreater than 1. The map [0 , × L n (0) → U n (0)( t, Γ) t − Γ(with the convention ∞ Γ = Γ , here Γ = 0) is continuous and induces a homeomor-phism cL n (0) → U n (0).Since U n (0) r { } ≃ (0 , × L n (0) is open, it inherits a stratification from that of C ( R n ). It follows that the intersections of L n (0) with the strata C ( p,q ) does definea stratification. The above homeomorphism becomes a stratified isomorphism when L n (0) is given this stratification.Note that we could do the same with any fixed value for the first norm instead of 1.The local study of the total group follows immediately from that of 0. Lemma 4.3 . —
The total group R n ∈ C ( R n ) has a neighborhood cL n ( R n ) where L n ( R n ) ≃ L n (0) is the set of subgroups of n -th norm . HE SPACE OF CLOSED SUBGROUPS OF R n Proof . — The duality map ∗ is a stratified isomorphism and maps 0 to R n . It musttherefore map U n (0) onto a neighborhood of R n . We can also reproduce the proof ofLemma 4.2: the neighborhood U n ( R n ) is the set of elements of n -th norm at most 1and the map [0 , × L n ( R n ) → U n ( R n )( t, Γ) t Γ(with the convention 0Γ = h Γ i , here h Γ i = R n ) induces an isomorphism cL n ( R n ) → U n ( R n ). Lemma 4.4 . —
Any type ( p, q ) element Γ ∈ C ( R n ) has a neighborhood of the form R ( n − p )( p + q ) × cL where the link L = L n (Γ ) is defined in U nδ (Γ ) (where δ is anysmall enough scale) by the equations ( V , V , V ) = ( V , V , V )Γ = Γ D N p (Γ ) + N p + q +1 (Γ ) − = δ/ where we use the notations of Section 4.2.3. Implicitely, L is stratified by its intersection with the strata of C ( R n ). Proof . — By homogeneity of strata we can restrict to Γ = R p × Z q . Consider itsneighborhood U nδ ( R p × Z q ): an element Γ there has a linear decomposition at scale δ ( V , V , V ) and a local decomposition (Γ , Γ , Γ , Φ , Φ ). Its projection to C ( p,q ) isdefined as V + τ − (Γ ) ⊂ V + V . It can be arbitrary in a neighborhood of R p × Z q in C ( p,q ) . As a consequence, U nδ ( R p × Z q ) is isomorphic the product of two sets, theset of possible choice of ( V , V , V , Γ ), which is a ( n − p )( p + q )-dimensional ball, andthe set M of possible choices of (Γ , Γ , Φ , Φ ). This last set is of course stratified bythe type ( r, s ) of the corresponding point Γ. This type depends only upon (Γ , Γ ).Apart from the choice of Φ and Φ , M looks like the product of two cones cL p ( R p )and cL n − ( p + q ) (0), and we proceed as in the proof of Lemma 3.6. The map[0 , × L n ( R p × Z q ) → M ( t, Γ , Γ , Φ , Φ ) ( t Γ , t − Γ , t Φ , Φ t )(where Φ t ( t − γ ) := ( t Φ ′ ( γ ) , Φ ′′ ( γ )) for all γ ∈ Γ ) induces the required isomorphism cL n ( R p × Z q ) → M .Note that Lemmas 4.2, 4.3 are included in this result. We now can tell that C ( R n )is Siebenmann stratified, but we can get more. Lemma 4.5 . —
Let Γ ∈ C ( R n ) and consider an element Γ ∈ C ( R n ) that lies onthe link L n (Γ ) . For small enough δ , the neighborhood U nδ (Γ) ∩ L n (Γ ) of Γ in L n (Γ ) is of the form R k × cL n (Γ) (where k depends on the types of Γ and Γ ).Proof . — This follows directly from previous lemma. Let ( p, q ) and ( r, s ) be the typesof Γ and Γ. Up to a change of scale cL n (Γ) ⊂ L n (Γ ) and L n (Γ ) intersects the ( r, s )strata of U nδ (Γ) along a submanifold of R ( n − r )( r + s ) . BENOˆIT KLOECKNER
The following last lemma settles the proof of Theorem B and shows how the generallinks are related to the L k (0). Lemma 4.6 . —
For all ( p, q ) , the link L = L n ( R p × Z q ) is a Goresky-MacPhersonstratified space.Moreover, if ( p, q ) is different from (0 , and ( n, , then the map π = π ( n, p, q ) : L = L n ( R p × Z q ) → L p ( R p ) ⋆ L n − ( p + q ) (0)(Γ , Γ , Φ , Φ ) (cid:18) N p (Γ ) − · Γ ; N p + q +1 (Γ ) · Γ ; 2 δ N p (Γ ) (cid:19) is a stratified bundle and for all ( r, s ) > ( p, q ) the fiber over the stratum L ( r,s ) is atorus of dimension q ( p − r ) + ( r + s − p − q )( p + q − r ) Proof . — The proof of the first part is by decreasing induction on ( p, q ), with respectto the usual ordering obtained from the condition of frontier.If ( p, q ) = (0 , n ), the link L n ( Z n ) is empty. If ( p, q ) < (0 , n ), Lemma 4.5 showsthat the link L = L n ( R p × Z q ) is a cone-like TOP stratified space with links of theform L n ( R r × Z s ) with ( r, s ) > ( p, q ), which are Goresky-MacPherson stratified byinduction hypothesis.For the second part, first remark that the map π restricts to a (classical) fiberbundle on each strata, and the fibers corresponds to the choice of Φ and Φ whengiven Γ and Γ . Each of these maps is defined by the image in a torus (of respectivedimension p − r and p + q − r ) of a basis of a lattice (of respective rank q and( r + s − p − q )); this gives the claimed topology for the fibers.Next we proceed by a similar induction than above. If ( p, q ) (0 , n ) and (Γ , Γ , λ )is a point in L p ( R p ) ⋆ L n − ( p + q ) (0), then ( r, s ) > ( p, q ) where r is the dimension ofthe continuous part of Γ and r + s − p − q is the rank of Γ . If U is a small enoughneighborhood of (Γ , Γ , λ ) and V = π − ( U ), then the restriction of π to V → U writes in the form required by Definition 3.7 with π ′ = π ( n, r, s ), thus is a stratifiedbundle by induction hypothesis.Note that the singular codimension of C ( R n ) is n , thus it is a pseudo-manifold if n >
2: the proof of Theorem B is over.The following will be central in the proof of Theorem C (recall that R m is thesubset of rank n element of C ( R n ) and R ℓ is its complement). Corollary 4.7 . —
Any type ( p, q ) element Γ ∈ C ( R n ) has a neighborhood system ( U ε ) ε such that U ε is contractible and U ε r R ℓ is pathwise connected.Proof . — First, the links L n (0) are pathwise connected. Indeed L n (0) ∩ R m is a densestrata, so that any point in L n (0) can be connected to a point in L n (0) ∩ R m , whichis pathwise connected (homeomorphic to GL( n ; R ) / GL( n ; Z )).Taking a neighborhood of Γ isomorphic to U = R ( n − p )( p + q ) × cL where L = L n ( R p × Z q ) and for U ε ( ε ∈ (0 , ε ball in R ( n − p )( p + q ) by the part { ( t, l ) ∈ cL | t < ε } of the cone cL , we get a neighborhood system such that U ε is HE SPACE OF CLOSED SUBGROUPS OF R n contractible and U ε r R ℓ is a deformation retract of the total space of a stratifiedbundle with tori as fibers and base L p (0) ⋆ (cid:0) L n − ( p + q ) (0) (cid:1) (0 ,n − ( p + q )) The tori are pathwise connected as well as L p (0). Moreover ( L n − ( p + q ) (0) (cid:1) (0 ,n − ( p + q )) is the set of unit norm lattices in R n − ( p + q ) and is therefore pathwise connected. Thepathwise connectedness of U ε r R ℓ follows. Let us consider some explicitexamples. We use Lemmas 4.2, 4.3 and 4.6. First, as already noticed, L (0) ≃ { Z } is reduced to a point and the case of the open strata is trivial: L n ( Z n ) = ∅ for all n . n = 2. — We already said that L (0) ≃ L ( R ) ≃ S where S is stratified with strata a trefoil knot and its complement. The proof iscontained in that of the Hubbard-Pourezza theorem, see the last section.The duality maps type (1 ,
1) elements to type (0 ,
1) ones, so that we have left toconsider only L ( Z ) and L ( R ). The link L ( Z ) is isomorphic to the set of couples(Γ , Φ ) where Γ ∈ L (0) ≃ { Z } and Φ is a homomorphism Γ → R / Z . As aconsequence, L ( Z ) ≃ S where S is stratified with one strata.The link L ( R ) is isomorphic to the set of triples (Γ , Γ , Φ ) where Γ ∈ U ε ( R )is defined by its norm α , Γ ∈ U ε (0) is defined by the inverse β of its norm, Φ is ahomomorphism from Γ ≃ β − Z to R / Γ ≃ R /α Z , and moreover α + β is constrainedto be equal to a constant δ/
2. As a consequence, L ( R ) ≃ S where S is stratified with three strata, two of them being reduced to a point (seefigure 8). L n ( Z n − ). — The link L ( Z ) is isomorphic to the set ofcouples (Γ , Φ ) such that Γ ∈ L (0) ≃ { } and Φ is a homomorphism from Z to R / Z . Therefore L ( Z ) ≃ T where T is the 2-torus stratified with a single stratum. This case is important,since it shows very simply that C ( R ) is not a manifold: Z has a neighborhoodhomeomorphic to R × cT . The same argument shows L n ( Z n − ) ≃ T n − BENOˆIT KLOECKNER Γ = 0Γ = R S ≃ { Φ : Γ ≃ Z → R / Z ≃ R / Γ } Figure 8.
The link L ( R ). thus the same conclusion holds for all n >
2. We see that C ( R ) is a manifold onlybecause of a “happy accident”: the cone over T is a 2-ball. C ( R ). — Let us give a few more examples with-out details. We have L ( R × Z ) ≃ T × T × T × [0 , / ∼ where the quotient is by the relations ( x, y, z, ∼ ( x ′ , y ′ , z,
0) and ( x, y, z, ∼ ( x, y ′ , z ′ , L ( Z ) → S where S is stratified by a trefoil knot and its complement, the fibers of this bundlebeing T (over generic points) and T (over singular points).There is a stratified bundle L ( R ) → ¯ cS where S is again stratified by a trefoil knot and its complement and ¯ cS = {•} ⋆ S is the closed cone over S , the fibers of this bundle being T (over generic points), T (over K × (0 , S × { } ).
5. Localization and simple connectedness
In this section we prove our main result. We start with the localization theoremD and then prove that C ( R n ) is simply connected. Let X be a Hausdorff topological space, Y be a closed subsetof X and m be any non-negative integer. Assume that each point y ∈ Y has anneighborhood system ( U ε ) in X such that the topological pair ( U ε , U ε r Y ) is m -connected. Let us prove by induction on k m that the pair ( X, X r Y ) is k -connected. In fact, we shall prove a stronger property to run the induction.For all k m , we denote by I k the cube [0 , k and by ∂I k its boundary, while 0denotes the point (0 , , . . . , x ∈ X r Y and let us prove by induction on k that any map α : ( I k , ∂I k , → ( X, X r Y, x ) is homotopic (with fixed boundary) to a map α : HE SPACE OF CLOSED SUBGROUPS OF R n I k → X r Y through a arbitrarily small homotopy ( α t ). More precisely, we shall provethat for any compact subsets K , . . . , K ℓ of I k and any open subsets W , . . . , W ℓ of X such that α ( K j ) ⊂ W j , we can ask that for all t , α t ( K j ) ⊂ W j .For any point x in the interior of I k , an open box around x is a neighborhoodof x that writes I × I × · · · × I k where I i are open intervals of [0 ,
1] that containneither 0 nor 1. The lower corner of an open box ] a , b [ × · · · × ] a k , b k [ is the point( a , a , . . . , a k ).We start with the case k = 0. We have to prove that any point in X is connectedby an arbitrarily small path to a point in X r Y . This follows directly from thehypothesis that U ε is a neighborhood system and ( U ε , U ε r Y ) is 0-connected.Assume now that we proved the desired result for maps ( I k − , ∂I k − , → ( X, X r Y, x ) and let α , ( K j ) and ( W j ) be as above.Denote by Σ := α − ( Y ) the singular set . It is a closed subset of I k , thus is compact.For all s ∈ Σ, there is a neighborhood U ( s ) of α ( s ) that is pathwise connected andsuch that ( U ( s ) , U ( s ) r Y ) is m -connected. When s ∈ K j , we can moreover assumethat U ( s ) ⊂ W j . Let V ( s ) = α − ( U ( s )) and B ( s ) be an open box around s such that B ( s ) ⊂ V ( s ). If s / ∈ K j , we moreover assume that B ( s ) ∩ K j = ∅ .Since Σ is compact, there exist a finite number of points s , . . . , s N ∈ Σ such thatthe B i := B ( s i ) cover Σ. Let us prove that α is homotopic to a map α for which N can be reduced. This will prove the theorem by induction on N , since N = 0 meansthat α avoids Y .Up to a reordering, we can assume that B has its lower corner x outside all ofthe B i . In particular, x / ∈ Σ. The restriction β of α to the boundary of B defines aelement in π k − ( U , x ) where U := U ( s ). Since ( U , U r Y ) is m -connected, β ishomotopic to a map β : ∂B → U r Y . The induction hypothesis moreover enablesus to assume that the homotopy ( β t ) is small enough to ensure β t ( K j ∩ ∂B ) ⊂ W j and β t ( ∂B r N [ i =2 B i ) ⊂ X r Y for all t . In particular, the homotopy β t will not add any new singular part at thenext step.Composing as in figure 9 the part of α exterior to B with the homotopy from β to β ′ , then with its inverse, finally with the restriction of α to B , we can assume that α maps ∂B in U r Y while ensuring that Σ is still covered by the B i .Now the restriction of α to B defines an element of π k ( U , U r Y, x ) and there isa homotopy H : [0 , × ( B , ∂B , x ) → ( U , U r Y, α ( x )) such that H (0 , · ) = α | B and H (1 , · ) takes its values in U r Y . This homotopy extends to a homotopy between α and a map α whose singular set Σ is covered by the N − B , B , . . . , B N .Moreover, our assumptions ensure that for all j , either B ∩ K j = ∅ or U ⊂ W j ,therefore our homotopy is small enough to carry the induction. This finishes theproof. We note C = C ( R n ) and recall that R m is theset of closed subgroups of maximal rank and R ℓ is its complement. BENOˆIT KLOECKNER (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) αβ B Σ βαβ β B α Figure 9.
On the left, dashed lines represent boxes B i for i >
1. On theright, in the gray region we composed α with the homotopy from β to β and back to β . Lemma 5.1 . —
For each Γ ∈ R ℓ , if we denote by ( U ε ) the neighborhood systemgiven in Corollary 4.7, the pair ( U ε , U ε r R ℓ ) is -connected.Proof . — We know that U ε is contractible, thus pathwise connected and simply con-nected. The pair ( U ε , U ε r R ℓ ) is in particular 0-connected. Moreover U ε r R ℓ ispathwise connected. But we have an exact sequence1 = π ( U ε ) → π ( U ε , U ε r R ℓ ) → π ( U ε r R ℓ ) = 1thus π ( U ε , U ε r R ℓ ) is trivial, as desired.This classical exact sequence is very easy to understand it this case: any curve in U ε whose ends lie in U ε r R ℓ is homotopic to a curve whose ends coincide and lie in U ε r R ℓ , simply because this set is arc-connected. But since U ε is simply connected,this curve is nullhomotopic, thus homotopic to a curve entirely lying in U ε r R ℓ .We can now complete the proof of Theorem C. Since R ℓ is the closure of the strataof type (0 , n −
1) and thanks to the preceding lemma, the localization theorem impliesthat ( C , R m ) is simply connected. This means that any loop of C based at R n ishomotopic to a loop in R m .The map defined on R m × [0 ,
1] by H (Γ , t ) = t Γ is a continuous homotopy betweenthe constant map with value R n and the identity map. Therefore, any loop of C is nullhomotopic. Note that the extension of H on the whole of C would not becontinuous at t = 0, since it fixes 0 but retracts lattices of arbitrarily large norm to R n .We cannot prove this way that C is 2-connected. We would indeed need the2-connectedness of ( U ε , U ε r R ℓ ) which does not hold. For example, a typical neigh-borhood for Z n − in R n has the homotopy type of the cone over a ( n − HE SPACE OF CLOSED SUBGROUPS OF R n intersection with R ℓ being the apex. The torus is not simply connected, thus the pair( cT n − , T n − ) is not 2-connected.
6. The Chabauty space of R is a -sphere6.1. Definitions and notations. — In this section, we denote by C the Chabautyspace of R . A closed subgroup of R is of one of the following types: – (0 , – (0 , Z ; – (0 , Z (these are the lattices) ; – (1 , R ; – (1 , R × Z ; – (2 , R .Each type is an orbit of the action of GL(2; R ) on C . The set of lattices is L := C (0 , ,its complement is denoted by H .A closed subgroup Γ of R has a determinant, or covolume, covol(Γ). If Γ is alattice, it is its usual determinant, that is the determinant of any direct base of Γ. Itis 0 if Γ is isomorphic to R × Z or R , and ∞ if Γ is isomorphic to Z or 0. In otherwords, it is the 2-dimensional volume of the quotient R / Γ. By convention, covol(Γ)takes simultaneously all values in [0 , ∞ ] if Γ is isomorphic to R . So defined, the levelsof covol are closed in C . Outside the set R := C (1 , of such subgroups, covol is acontinuous function. Beware that here, the letter R refers to R (and not to the rank).Let C > , respectively C , be the subsets of C defined by covol > R . Let H > = H ∩ C > be the set of subgroups isomorphicto R , Z or 0, and H = H ∩ C be the set of subgroups isomorphic to R , R × Z or R .Let L be the set of covolume 1 lattices, and C its closure. Then C is the unionof L and of the set R .We use the usual identification R ≃ C , so that any subgroup isomorphic to R canbe written in the form e iθ R .We also define as before the norm (or systol) N (Γ) = N (Γ) = inf {| x | | x ∈ Γ \ { }} It is a continuous functions taking its values in [0 , ∞ ]. Let C be the set of norm 1subgroups of R . A point of C is either isomorphic to Z , or a lattice. We denote by Z the set C \ L .Figure 10 sums up this notations.The proof of Theorem A is in two parts. We first prove that the topological pair( C , H ) is the suspension of ( C , Z ), then that the latter is homeomorphic to ( S , K )where K is a trefoil knot. R is a suspension. — Lemma 6.1 . —
The topological pair ( C > , H > ) is homeomorphic to the cone over ( C , R ) . BENOˆIT KLOECKNER C > H R = C (1 , C Z H > C R C Figure 10.
Sum up of notations
Proof . — We consider the mapΦ : C × [0 , ∞ ] → C > (Γ , t ) ( (cid:16) tN (Γ ) + 1 (cid:17) Γ if Γ ∈ L te iθ Z if Γ = e iθ R where by convention 0 e iθ Z = e iθ R and ∞ Γ = 0 if Γ is discrete.This map is continuous, maps C × { } onto C and R × [0 , ∞ ] onto H > . Itinduces a continuous bijection ˜Φ from the quotient of C × [0 , ∞ ] by the relation(Γ , ∞ ) ∼ (Γ ′ , ∞ ) onto C > . Since the latter is compact, ˜Φ is a homeomorphismbetween the cone over ( C , R ) and ( C > , H > ). Lemma 6.2 . —
The topological pair ( C , R ) is homeomorphic to ( C , Z ) .Proof . — The map Ψ : C → C that assigns to Γ the only t Γ of unit covolume ( t = 0if Γ is isomorphic to Z , t = covol(Γ) − / otherwise) is continuous and a bijection. Bycompacity of C , closed in C , it is a homeomorphism. Proposition 6.3 . —
The topological pair ( C , H ) is homeomorphic to the suspensionof ( C , Z ) .Proof . — We can either reproduce the previous arguments to prove that ( C , H )is also a cone over ( C , Z ) or use the duality ∗ which maps C > on C and preserves L . To get Theorem A, we have left to prove thefollowing.
Proposition 6.4 . —
The topological pair ( C , Z ) is homeomorphic to ( S , K ) . HE SPACE OF CLOSED SUBGROUPS OF R n The proof runs over the rest of the Section. We shall describe C as a Seifertfibration. Let Γ be a point of C . The isometry group SO(2) acts on C , and up toa rotation we can assume that 1 ∈ Γ ⊂ C . Then Γ is determined by the choice of asecond vector in the fundamental domain D = { z ∈ C ; | z | > − / > Re( z ) > / } ∪ {∞} where z = ∞ means that Γ is isomorphic to Z (figure 11). Identifying the points of D that represent the same Γ leads to the quotient of D by the relation z ∼ z − z ) = 1 / z ∼ − ¯ z if | z | = 1, turning it into a 2-sphere denoted by B , that willbe the base of the Seifert fibration. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Figure 11.
Fundamental domain: the vertical lines and the circle arcs areglued according to the arrows, (cid:3) and △ are the singular points. The kernel of the action of SO(2) is reduced to {± } , and the quotient gives anaction of the circle that is almost free: the only points of C that have nontrivialstabilizers are the triangular lattices (stabilizer of order 3) and the square lattices(stabilizer of order 2). It follows that C is a Seifert fibration with base B ≃ S and two singular fibers of order 2 and 3, and where Z is a regular fiber. Theunnormalized Seifert invariants of C are (0 | (2 , β ); (3 , β )) and we have left to findthe rational Euler number β / β / C , Z ).We first choose a cross-section of the regular part of the Seifert fibration. It wouldbe natural to lift each point u in the fundamental domain to the subgroup generatedby u and 1, but this would not define a continuous cross-section. The gluing ofthe unit circle indeed identifies, for all θ ∈ [0 , π/ Z + e i ( π/ − θ ) Z and 1 Z + e i ( π/ θ ) Z by a rotation of angle π/ θ . We shall therefore modify thiscross-section in a neighborhood of one of the circular arcs of D .Let S = R /π Z be the quotient SO(2) / {± } , D ′ be the fundamental domain D minus the singular points ( i , e iπ/ and e iπ/ ) and B ′ be the base B minus the twosingular points (corresponding to i and e iπ/ ∼ e iπ/ ). We choose a continuous map f : D ′ → [0 , π/
2] that is constant with value 0 except in a neighborhood of the arc (cid:8) e i ( π/ θ ) (cid:12)(cid:12) θ ∈ ]0 , π/ (cid:9) , where it satisfies f ( e i ( π/ θ ) ) = π/ − θ . We then define across-section σ : B ′ → C by σ ( u ) = e if ( u ) (1 Z + u Z ). It is continuous since σ ( e i ( π/ θ ) ) = e i ( π/ − θ ) (1 Z + e i ( π/ θ ) Z ) = e i ( π/ − θ ) Z + 1 Z = σ ( e i ( π/ − θ ) ) BENOˆIT KLOECKNER
Let b be the homotopy class in C of a regular fiber, d and d be the homotopyclasses defined by σ on the boundary of C • = C \ { F , F } where F and F areinvariant neighborhoods of the singular fibers of order 2 and 3, respectively. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) f = 0 d d Figure 12.
The cross-section σ defines homotopy classes in the boundaryof C • . In ∂F and ∂F respectively, we get that 2 d + b and 3 d − b are homotopic tomeridians (see figure 13 where F and F are pictured with coordinates ( u, ϕ ) ∈ B × R /π Z e iϕ (1 Z + u Z ), with the suitable identifications). It follows that C hasunnormalized Seifert invariants (0 | (2 , , (3 , − / − / / bd π π bd π π π Figure 13.
Neighborhood F and F of the singular fibers We shall know exhibit a very classical Seifert fibration on S whose regular fibersare trefoil knots, that has base S , two singular fibers of order 2 and 3 and rationalEuler number 1 /
6. Since a Seifert fibration is determined by these data, we willconclude that ( C , Z ) is homeomorphic to ( S , K ).Consider the following action of the circle R / Z on S , identified to the unit sphereof C : s · ( z , z ) = ( e πm is z , e πm is z )with m = 2 and m = 3. The stabilizer of almost every point is trivial, the exceptionsbeing the polar orbits ( z ,
0) and (0 , z ). If m and m where equal to 1, we would HE SPACE OF CLOSED SUBGROUPS OF R n get the Hopf fibration where the non-polar orbits are Villarceau circles of the tori | z /z | = c , where c runs over [0 , ∞ ]. Taking m = 2 and m = 3, we replaced theVillarceau circle by toric knots, here trefoil knots (figure 14). Figure 14.
The torus knot (2 ,
3) is a trefoil knot.
We see that the regular part of the base is foliated by the circles obtained byquotienting the tori | z /z | = c by the action of S , and is therefore an annulus. Onecan see this annulus as the S base of the Hopf fibration minus two points for thesingular fibers.Let us compute the Seifert invariants of this action, which are surprisingly difficultto find in the litterature. We use a representation found in [ ].Let T = R / Z × RZ be the standard 2-torus equiped with the foliation by straightlines of slope 3 /
2. If we denote by x the homotopy class of R / Z → T t ( t, y the homotopy class of R / Z → T t (0 , t )the homotopy class of any leave of this foliation is ℓ = 2 x + 3 y .In the space T × [0 ,
1] define T t := T × { t } , endowed with the above foliationfor t ∈ (0 , T × [0 , → S be the mapping defined as follows. First,Π contracts T to the singular fiber { (0 , z ) | | z | = 1 } and T to the singular fiber { ( z , | | z | = 1 } with Π( a, b,
0) = (0 , e iπb ) and Π( a, b,
1) = ( e iπa , T t to a torus defined by | z /z | = c ( t ) with c an increasing continuous functionsuch that c ( t ) → ∞ ) when t → T t tothe Seifert foliation in S . Think of T × [0 ,
1] as a blow-up of S along the singularfibers.The point is that in this presentation, one can give explicitely a cross-section ofthe Seifert fibration over the regular part: just consider the set { ( s, s, t ) | s ∈ R / Z , t ∈ (0 , } ⊂ T × (0 , BENOˆIT KLOECKNER
This set intersects each of the T t along a straight line homotopic to x + 2 y , whichintersects each 2 x + 3 y line once, thus it does define a section.In the boundary of a neighborhood of T , the section defines a curve homotopic to d = − x − y (the sign depends upon the choice of orientation). Since ℓ = 2 x + 3 y is the homotopy class of a regular fiber, we have 3 d + 2 ℓ = x , a meridian. Similarly,in the boundary of a neighborhood of T , the section defines a curve homotopic to d = x + 2 y and 2 d − ℓ = y is a meridian.Therefore, this Seifert fibration has unnormalized invariants (0 | (3 , , (2 , − / − / / Remark 6.5 . — It is well known (see e.g. [ ]) that SL(2; R ) / SL(2; Z ) is homeo-morphic to the complement of a trefoil knot in S . This can be given an alternativeproof using the same methods we used to prove Theorem A. Lemma 6.2 indeed showsthat this homogeneous space is homeomorphic to C \ Z . The study of the Seifertfibration on C implies this result, but the study of C \ Z is in fact simpler. Moreprecisely, we do not need to study how the singular fibers are glued to the regularpart, the non-compacity enabling one to assume β = 1 and β = 1 or 2. Moreover,the difference between β = 1 and β = 2 ( ≡ − R ) / SL(2; Z ), we obtain the complement of a right or left trefoil knot). Remark 6.6 . — Christopher Tuffley studied [ ] the spaces exp k ( S ) of all non-empty subset of the circle of cardinality at most k . In particular, he proved usingSeifert fibrations that exp ( S ) is a 3-sphere, its subset exp ( S ) being a trefoil knot.The similarity with Proposition 6.4 is not fortuitous: Jacob Mostovoy proved[ ] by a simple geometric argument that (exp ( S ) , exp ( S )) is homeomorphic to( C , Z ). Combining these two results one gets another Seifert fibration proof ofProposition 6.4. Note that even the Seifert part is somewhat different from ours,since it is first proved that exp ( S ) is simply connected, which reduces drasticallyits possible Euler numbers. Remark 6.7 . — A nice feature of the study of exp ( S ) is that its subset exp ( S )is easily seen to be a M¨obius strip, with boundary exp ( S ): we recover the fact thata trefoil knot bounds a M¨obius strip. This can be seen in ( C , Z ) as well: over thevertical line L = { iy | y ∈ [1 , + ∞ ] } of the base B , the Seifert fibration is a closedM¨obius strip whith boundary Z , obtained by identifying antipodal points of the( y = 1) boundary component of the strip L × S .
7. A few open questions
There are many question left open concerning C ( R n ). Let us consider some ofthem that seem of special interest.1. Determine whether C ( R n ) is stratified in the sense of Thom or Mather. Itwould for example imply that it can be triangulated ([
11, 8 ]). More ambitiously,determine if we can endow C ( R n ) with the structure of an algebraic variety. This HE SPACE OF CLOSED SUBGROUPS OF R n question is motivated by the original proof of the Hubbard-Pourezza theorem,where the link L (0) is described by algebraic means.2. Compute the intersection homology of C ( R n ).3. Describe explicitely C ( R ), or at least the set L (0) of unit norm subgroups of R . References [1]
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