TTHE WEAK B-PRINCIPLE: MUMFORD CONJECTURE
RUSTAM SADYKOV
Abstract.
In this note we introduce and study a new class of mapscalled oriented colored broken submersions. This is the simplest classof maps that satisfies a version of the b-principle and in dimension 2approximates the class of oriented submersions well in the sense thatevery oriented colored broken submersion of dimension 2 to a closedsimply connected manifold is bordant to a submersion.We show that the Madsen-Weiss theorem (the standard MumfordConjecture) fits a general setting of the b-principle. Namely, a version ofthe b-principle for oriented colored broken submersions together with theHarer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem. Introduction
A smooth map of manifolds f : M → N is said to be an immersion ifits differential is a fiberwise monomorphism T M → T N of tangent bun-dles. According to a remarkable theorem by Smale and Hirsch the spaceof immersions M → N of given manifolds with dim M < dim N is weaklyhomotopy equivalent to a simpler topological space of formal immersions ,i.e., fiberwise monomorphisms T M → T N . The Smale-Hirsch theorem wasone of the primary motivations for the general Gromov h-principle : given adifferential relation, the space of its solutions is weakly homotopy equivalentto the space of its formal solutions [9].In [14] (for a short review, see [15]) I proposed a stable homotopy versionof the h-principle, the b-principle, motivated by a series of earlier resultsincluding [1, 2, 5, 7, 11, 13, 16, 17, 19, 20]. Namely, with every open stabledifferential relation R , there are associated a moduli space M R of solutions,a moduli space h M R of stable formal solutions, and a map α : M R → h M R .It turns out that M R is an H-space with a coherent operation, while h M R isan infinite loop space [14], whose stable homotopy type is relatively simple.The b-principle is the following conjecture. The b-principle.
The canonical map M R → h M R is a group completion. When holds true, the b-principle allows us to perform explicit computa-tions of invariants of solutions. On the other hand, the b-principle is truefor most of the differential relations (see [14] and references above); notableexceptions are the differential relations of oriented submersions of positive
Mathematics Subject Classification.
Primary: 55N20; Secondary: 53C23. a r X i v : . [ m a t h . A T ] J a n RUSTAM SADYKOV dimensions d . In this important exceptional case the b-principle inclusioncoincides with the Madsen-Tillmann map α : (cid:116) BDiff M → Ω ∞ MTSO( d ) , where (cid:116) BDiff M is the disjoint union of the classifying spaces of orienta-tion preserving diffeomorphism groups of oriented closed (possibly not pathconnected) manifolds M of dimension d , while Ω ∞ MTSO( d ) is the infiniteloop space of the Madsen-Tillmann spectrum [7]. The standard Mumfordconjecture asserts that for d = 2 and for a closed oriented surface F g of genus g , the map α | BDiff F g induces an isomorphism of rational cohomology ringsin stable range of dimensions ∗ (cid:28) g . The Mumford conjecture was provedin the positive by Madsen and Weiss in [11], and later several other proofsof the Madsen-Weiss theorem were given in [7, 4, 8, 10]. Theorem 1.1 (Madsen-Weiss) . The rational cohomology ring of
BDiff F g is a polynomial ring in terms of Miller-Morita-Mumford classes κ i : H ∗ (BDiff F g ; Q ) (cid:39) Q [ κ , κ , ... ] , for ∗ (cid:28) g, or, equivalently, the map α | BDiff F g is a rational homology equivalence in astable range of dimensions.Remark . In fact, Madsen and Weiss proved a stronger statement, which,in particular, implies that the map α | BDiff F g is an integral homology equiv-alence in a stable range.In the current note we study a new class of flexible maps—the class ofcolored broken submersions—that provides a good approximation to theclass of submersions, retains the sheaf property, and satisfies a version of theb-principle. More generally, we define colored broken solutions to an openstable differential relation; these enjoy many interesting properties includingthe following ones. • For an open stable differential relation R that does not satisfy theb-principle, a stable formal solution of R can be integrated into abroken solution (Theorem 8.3). Thus, stable formal solutions dif-fer from solutions only in broken components of the correspondingbroken solutions. • The pullback of a colored broken solution with respect to a genericsmooth map is a colored broken solution. Thus, colored brokensolutions form a class and therefore possess a moduli space ( § • The class of colored broken solutions satisfies the sheaf property, andtherefore it is suitable for study by means of homotopy theory. • Colored broken solutions of an open stable differential relation R satisfy a weak b-principle (Theorem 8.3) even if solutions of R donot.To begin with we introduce the broken submersions/solutions in section §
2. In sections § HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 3 of submersions ( § § § § Theorem 1.3.
Let f : M → N be an oriented broken submersion of dimen-sion to a simply connected manifold N . Suppose that the image of brokencomponents of f in N is disjoint from ∂N ; in particular, over ∂N the map f is a fiber bundle with fiber F g . Suppose that g (cid:29) dim N . Then f is bordantto a fiber bundle by bordism which is a broken submersion itself. Theorem 1.3 relies heavily on the Harer stability theorem, and its proofis very much in spirit of a singularity theoretic argument by Eliashberg,Galatius and Mishachev in [4].Next we review the weak b-principle ( § § M b of colored broken submersionsis an H-space with coherent operation. Its classifying space B M b is known;it has essentially been determined in [7] (for a proof in present terms, see[14]). Finally, in section 10 we show that in view of the Harer stabilitytheorem and the Miller-Morita theorem, the Madsen-Weiss theorem followsfrom the weak b-principle for colored broken submersions.Colored broken submersions are similar to (but have better propertiesthan) marked fold maps. In particular, the moduli space of colored brokensubmersions of dimension d is an appropriate homotopy colimit of classifyingspaces BDiff M of diffeomorphism groups of manifolds of dimension d withcertain boundary components, compare with the original paper [11]. Coloredbroken submersions should be compared with enriched fold maps from [4] ofGalatius-Eliashberg-Michachev who used them to give a topological proofof the Madsen-Weiss theorem. Note, however, that in contrast to enrichedfold maps, colored broken submersions behave well with respect to takingpullbacks and possess a moduli space ( § B M b isessentially from [7] (however, the rest of their proof of the Madsen-Weisstheorem is not necessary in the current setting). Acknowledgement.
I am grateful to Soren Galatius for his generous help;the key idea to use I -spaces (e.g., see [18]) to link the b-principle to theMumford conjecture is his. I would also like to thank Ivan Mart´ın Protossfor presenting the material of the note in a series of talks in Topology Seminarin CINVESTAV. This paper was partially written while I was staying at theMax Planck Institute for Mathematics.2. Broken solutions
Given a smooth map f : M → N , a point x ∈ M is said to be regular ifin a neighborhood U of x the map f | U is a submersion. A point x ∈ M isa fold point if there are coordinate charts about x and f ( x ) such that(1) f ( x , ..., x m ) = ( x , ..., x n − , ± x n ± x n +1 ± · · · ± x m ) , RUSTAM SADYKOV where n is the dimension of N , and x , ..., x m are coordinates in the coordi-nate chart about x . If every point in M is regular or fold, then f is said to bea fold map . It immediately follows from the local coordinate representation(1) of f that the set of fold points of f is a submanifold of M of codimension d + 1 where d = dim M − dim N . Figure 1.
A breaking component.Suppose that a path component σ of fold points of f is closed in M andthe restriction f | σ is an embedding. Suppose that there is a submersion τ M of a neighborhood of σ in M onto a neighborhood of 0 in R d +1 such that theinverse image of 0 is precisely σ . Then the map τ M trivializes the normalbundle of σ , though we do not fix a diffeomorphism of a neighborhood of σ onto σ × R d +1 . Similarly, suppose that there is a map τ N of a neighborhoodof f ( σ ) to R trivializing the normal bundle of f ( σ ) in such a way that τ N ◦ f = g σ ◦ τ M on the common domain, where g σ is a Morse function on R d +1 with one critical point. Then we say that σ is a broken component;the maps τ M and τ N are parts of the structure of a broken component. Theminimum of the indices of the critical points of g σ and − g σ is called the index of σ . Remark . The normal bundle in M of a component σ of fold points of ageneral fold map f is not trivial, and f | σ is not necessarily an embedding.Therefore not every component of fold points of a fold map admits a struc-ture of a broken component. In fact, even if f | σ is an embedding and thenormal bundles of σ in M and f ( σ ) in N are trivial, the component σ maystill not admit a structure of a breaking component since f near σ may betwisted. Remark . Broken components of index 0 are not compatible with certainnice structures including the structure of broken Lefschetz fibrations in thecase of maps of 4-manifolds into surfaces. For this reason in the generalsetting in [15] we prohibited broken components of index 0 and proved theweak b-principle in the form of Theorem 8.3 with a less restrictive assump-tion of indices (cid:54) = 0. For the argument in the present paper, however, it isconvenient to allow broken components of index 0 (so that the space M b in § HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 5
Given an open stable differential relation R imposed on maps of dimension d , suppose a map f away of the broken fold components is a solution. Thenwe say that f is a broken solution of R .3. Bordisms
We need the notion of an oriented bordism of maps of manifolds withboundaries. An oriented bordism of a manifold with boundary is an orientedbordism with support in the interior of the manifold. An oriented bordismof maps is defined appropriately.
Definition 3.1.
Let M be an oriented compact manifold with corners suchthat ∂M is the union of − M , M and ∂M × [0 ,
1] where ∂M × { i } and ∂M i are identified for i = 0 ,
1, see Figure 2. In particular, the manifolds ∂M and ∂M are canonically diffeomorphic. The corners of M are along ∂M × { i } . Let N be an oriented compact manifold with corners and with Figure 2.
An oriented bordism.a similar decomposition of the boundary. Let f : M → N be a map thatpreserves the decompositions. In particular, appropriate restrictions of f define two maps f i : ( M i , ∂M i ) → ( N i , ∂N i ) , where i = 0 , . We say that f is an oriented bordism from f to f if f = f i × id and f = f × id [0 , over collar neighborhoods of M i and ∂M × [0 ,
1] respectively.If f , f belong to some class of maps, then we require that f belongs to thesame class. For example, a bordism of fiber bundles is a fiber bundle.The product map F × id [0 , : M × [0 , → N × [0 ,
1] is said to be a trivial bordism. Let m ⊂ M be a compact submanifold of codimensionzero, and f : m → N × [0 ,
1] a bordism of f = F | m . Then, there is a well-defined bordism F : M → N × [0 ,
1] where M is obtained from M × [0 ,
1] byremoving m × [0 ,
1] and attaching m along the new fiberwise boundary. Themap F coincides with f over m and with F × id [0 , over the complementto m . We say that F is a bordism of F with support in m and with core f . RUSTAM SADYKOV Concordances
A bordism M → N of maps is said to be a concordance if the manifold N is a product N × [0 , N = N × { } . Thus, for example, two proper maps f i : M i → N with i = 0 , concordant if there is a proper map f : M → N × [0 ,
1] together withdiffeomorphisms f − ( N × [0 , ε )) ≈ M × [0 , ε ) , f − ( N × (1 − ε, ≈ M × (1 − ε, ε -neighborhoods of M and M for some ε > f | f − ( N × [0 , ε )) = f × id [0 ,ε ) , f | f − ( N × (1 − ε, f × id (1 − ε, , see Figure 3. A concordance of maps of a given type is required to be a mapof the same type, e.g., a concordance of submersions is a submersion. Figure 3.
Concordance
Figure 4.
The map ( g, α )One concordance, called breaking , is of particular interest. It is con-structed by means of a compact manifold W of dimension d , and a properMorse function f on the interior of W with values in (0 , ∞ ). Suppose that f − [1 , ∞ ) is diffeomorphic to ∂W × [1 , ∞ ) and, furthermore, the restrictionof f to the latter is the projection onto [1 , ∞ ). Then( g, α ) : Int W × S n f × id −−−→ (0 , ∞ ) × S n ⊂ −→ R n +1 (cid:39) R n × R is a broken submersion [15], see Fugure 4. The inclusion (0 , ∞ ) × S n ⊂ R n +1 in the composition takes a scalar r and a vector v ∈ R n +1 of length 1 to rv .By [15, Proposition 4.2], the map g is also a fold map.Let i A denote the inclusion of a subset A into R , and let ( g A , α A ) denotethe pullback of the map ( g, α ) : Int W × S n → R n × R with respect to( i A × id R n − ) × id R : ( A × R n − ) × R −→ R n × R . Then ( g [0 , , α [0 , ) is a concordance, see Figure 5. Its inverse is a concor-dance from ( g , α ) to ( g , α ). It is called the standard model for breakingconcordances as this concordance breaks fibers of a submersion. HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 7
Figure 5.
Breaking concordanceFinally, for any map ( f, α ) : W → N × R and any of its regular points p ,there is a neighborhood U ≈ R d + n − × R of p in W such that ( f, α ) has theform ( g , α ) over U . We say that a concordance of ( f, α ) is breaking if itcoincides with the standard model for breaking concordances over U , and itis trivial elsewhere (i.e., it has support in U ).5. Basic concordances
We will show that Theorem 1.3 follows from the Harer stability theorem.The argument is in spirit of that by Eliashberg-Galatius-Mishachev in [4]. Inthis section we consider two basic concordances that will play an importantrole in the proof.
Example 5.1.
Let π : E → N be a fiber bundle with fiber a surface F g of genus g . Let D , D be two disjoint submanifolds of E such that π | D i is a trivial disc bundle over N . In particular, D i = N × D . We aim toconstruct a broken fold concordance of π to a fiber bundle with fiber F g +1 ,see Figure 6.Constant maps of D (cid:116) D and D × S to a point are concordant by meansof a Morse function u : W → [0 ,
1] with a unique critical point (of index 1),see Fig. 7. Let Π be the concordance of π with support in D (cid:116) D and withcore id × u : N × W → N × [0 , stabilizing concordance ; itattaches to each fiber F g a handle, see Figure 6.A stabilizing concordance also exists in a slightly more general settingwhere π : E → N is a broken fibration, and D , D two disjoint submanifoldsof E such that each π | D i is a trivial disc bundle over N .In general, however, a given fiber bundle π : E → N may not containtrivial disc subbundles. For this reason we also introduce a concordanceof Example 5.2 which stabilizes fibers locally, only over a subset U ⊂ N ;such a concordance always exists. First we will explain the constructionin the model case where N ⊂ R n is a disc and π is a disjoint union oftwo disc bundles, and then we consider the general case. The fibers of thisconcordance are presented on Figure 8. RUSTAM SADYKOV
Figure 6.
A stabilizing concordance. u Figure 7.
Cobordism W . Example 5.2.
For the construction we will need a compact manifold W ,and a proper Morse function h on the interior of W such that the fibers of h over negative and positive values are D (cid:116) D and D × S respectively,compare h with the function u on Figure 7.Let f be the disjoint union of two trivial disc bundles D i = D × N → N , i = 1 ,
2, over the standard open disc N ⊂ R n of radius 1. Let U ⊂ N be theconcentric closed subdisc of radius 0 .
5, see the part of Figure 8 over N × { } . Figure 8.
Fibers over N × [0 , S be the lower hemisphere of the sphere in N × [0 , ⊂ R n × R ofradius 0 . { }×{ } ; it meets the boundary N ×{ } transversallyalong ∂S = ∂U × { } and the projection of the interior of S to N × { } isa diffeomorphism onto the interior of U × { } , see Figure 8. We define f tobe the broken submersion to N × [0 , ⊂ R n +1 given by the restriction of W × S n h × id Sn −−−−−→ R × S n −→ R n +1 , where the second map in the composition takes a real number λ and a vector v of length 1 to λv + e n +1 . Thus, over a neighborhood S × R of S in N the concordance f is given by id S × h and over each path component of thecomplement to S it is a trivial fiber bundle.We will use this concordance in a more general setting.Let f be a broken submersions E → N and U ⊂ N a small disc withsmooth boundary. We aim to construct a concordance which attaches to HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 9 each fiber over the interior points of U a handle. We identify U with a closedball in R n of radius 0 .
5, and a neighborhood V of U in N with an open ballof radius 1. If U is sufficiently small, then E | f − V contains two disjointsubmanifolds D and D such that each f | D i is a trivial disc bundle over V .We have constructed the concordance of f | D (cid:116) D . Since it is trivial nearthe fiberwise boundary, we can extend the constructed concordance triviallyto a concordance of f | f − ( V ). Since the obtained concordance is trivialnear f − ( ∂V ), we may extend it trivially to a desired concordance of f .An important consequence of the concordance in Example 5.2 is the fol-lowing proposition. Proposition 5.3.
Let f : M → N be a broken submersion over a compactmanifold. If over (possibly empty) ∂N the original map f is a fiber bun-dle with fiber F g of genus g (cid:29) dim N , then f is concordant to a brokensubmersion f with connected fibers such that each regular fiber is of genus (cid:29) dim N . Folds of index
Erasing concordance.
Let F be an oriented closed surface, and N an arbitrary manifold. Then the broken submersion given by the projection N × F → N is concordant to an empty map. The concordance is given bya broken submersion of N × W where W is an oriented compact 3-manifoldwith ∂W = F . For example, if W is the standard 3-disc of radius 1 / √ erasing concordance id N × h where h ( x ) = −| x | + 0 . N with the empty map.6.2. Chopping concordance.
Let π : E → N be a submersion of dimen-sion 2 with fiber F g and D → N a trivial open disc subbundle of π . A chopping concordance chops off a sphere from each fiber. More precisely,a chopping concordance modifies the fiber bundle only inside D so we willassume that E = D . There are a bordism W from D to D (cid:116) S , and aMorse function f : W → [0 ,
1] with a unique critical point of index 2. Thedesired concordance is id N × f .The following proposition at least in part appears in [11] and [4]. Proposition 6.1.
Every proper broken submersion f of even dimension d to a compact simply connected manifold N is concordant to a broken sub-mersion f with no fold points of index .Proof. Suppose that N is closed. Let σ be a component of folds of f of index0, and let U denote one of the two path components of the complement to f ( σ ) in N for which the coorientation of f ( σ ) is outward directing. Theconcordance that we construct is trivial outside a neighborhood of f − ( ¯ U ).Consequently, we may assume that N is a neighborhood of ¯ U . In fact, onlythe component containting σ is modified, and therefore, by the definition of broken submersions, we may assume that M = σ × R d +1 , and f is theproduct of id σ and g = x + · · · + x d +1 followed by an identification of σ × R with a neighborhood of σ in N . Let S be a submanifold in N × [0 , ∂S = ∂ ¯ U × { } and the projection of the interior of S to N is adiffeomorphism onto U . Over a neighborhood S × R of S the map f is givenby id S × g , while over each of the two components of the complement to S in N × [0 , f is a trivial fiber bundle.Suppose now that N has a non-empty boundary. Let σ be a componentof folds of index 0. If f ( σ ) bounds S and the coorientation of ∂S is outwarddirecting, then σ can be eliminated by the concordance of the first part ofthe proof. Suppose ∂S is inward directing. Let N (cid:48) denote the enlargementof N with a collar ∂N × [0 ,
1] attached to N by means of an identificationof ∂N ⊂ N with ∂N × { } . Let’s extend f over the collar so that it is aconcordance that first chops off a sphere from each fiber and then eliminatesthe choped off component by the erasing concordance. In particular theextended map f has a new component σ (cid:48) of breaking folds of index 0.Furthermore, the image of σ (cid:48) (cid:116) σ bounds S (cid:48) ⊂ N (cid:48) such that the coorientationof ∂S (cid:48) is outward directing. Hence, σ and σ (cid:48) can be eliminated by theconcordance of the first part of the proof. Thus, we can assume that f hasno folds of index 0. (cid:3) Geometric consequences of the Harer stability theorem
Let Γ g,k denote the relative mapping class group of a surface F g,k of genus g with k boundary components. There are several proofs of the Mumfordconjecture, most of them use the Harer stability theorem: the homomor-phism Γ g,k → Γ g,k − induced by capping off a boundary component of F g,k and the homomorphism Γ g,k → Γ g +1 ,k − induced by attaching a cylinderalong two boundary components are homology isomorphisms in dimensions (cid:28) g . In view of the Atiyah-Hirzebruch spectral sequence, the Harer stabilitytheorem is equivalent to the assertion that the homomorphisms under con-sideration induce bordism isomorphisms of classifying spaces in dimensions (cid:28) g . Example 7.1.
By the Harer stability theorem, given a fiber bundle f : E → N over a compact manifold of dimension (cid:28) g with fiber F g,k and a section s over ∂N together with a trivialization τ of the normal bundle of s ( ∂N ) in E | ∂N , there are an oriented bordism of f to f : E → N and extensionsof s from ∂N = ∂N over N and τ from s ( ∂N ) over s ( N ). Indeed, theinitial data defines a map of pairs( N , ∂N ) → (BDiff F g,k , BDiff F g,k +1 ) , and the assertion is equivalent to the existence of a bordism to a map withimage in BDiff F g,k +1 . Example 7.2.
Let f be a fiber bundle over N with fiber F g of genus g (cid:29) dim N . Suppose that there exists a stabilization f of f , see Example 5.1. HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 11
Then f is zero bordant if and only if f is. Indeed, the assertion followsfrom the fact that the two inclusionsBDiff F g ←− BDiff F g, −→ BDiff F g +1 are bordism equivalences in stable range.Eliashberg, Galatius and Mishachev gave [4] an important geometric in-terpretation of the Harer stability theorem. In this section we deduce twoconsequences of the Harer stability theorem (Proposition 7.3 and 7.6) forbroken submersions using a singularity theory technique from [4]. Proposition 7.3.
Let f be a broken submersion M → N to a closedsimply connected manifold N . Then f is bordant to a fiber bundle.Proof. In view of Propositions 5.3 and 6.1, we may assume that the fiberof f over each regular point is a connected surface of genus (cid:29) dim N andthat f has no folds of index 0. Let σ denote a path component of breakingfolds Σ f of f . Since f ( σ ) is cooriented and N is simply connected, theMayer-Vietoris sequence implies that the complement to f ( σ ) consists of h Figure 9.
The image g ( ˜ S ).two components. Let S denote the closed submanifold in N bounded by f ( σ ) such that the coorientation of the fold values ∂S is inward directed,see Figure 9. Recall that a neighborhood of σ is identified with σ × R andnear σ the map f is given by id σ × m where m = − x − x + x . Let ˜ σ bethe submanifold σ × { } × { } × R in the neighborhood of σ . Note that thecoordinates x and x trivialize the normal bundle of ˜ σ . Let ˜ S = S ∪ ∂S S bethe double of S . A neighborhood ˜ σ (cid:48) of ∂S in ˜ S is canonically diffeomorphicto ˜ σ (cid:48) . Given a map h of ˜ S , the restrictions of h to the two copies of S aredenoted by h + and h − .In view of Lemma 7.4 below, we may assume that the canonical diffeo-morphism ˜ σ (cid:48) → ˜ σ extends to an inclusion h : ˜ S ⊂ M such that h + and h − are right inverses of f , and the trivialization of the normal bundle of ˜ σ extends to that over h ( ˜ S ).The promised concordance will have support in a small neighborhood h ( ˜ S ) × R of h ( ˜ S ); hence, we may assume that the complement is empty. Let S (cid:48) be a copy of S in N × [0 ,
1] such that S (cid:48) meets the boundary of N × [0 ,
1] transversally along ∂S × { } and the projection of the interior of S (cid:48) to N is a diffeomorphism onto the interior of S . Over a neighborhood S (cid:48) × ( − ,
1) of S (cid:48) in N × [0 , S (cid:48) × u , where u isthe Morse function of Example 5.1 (see Figure 7), while over the complementto S (cid:48) in N the concordance is trivial. (cid:3) Lemma 7.4.
After possibly modifying f by an oriented bordism, we mayassume that there is an embedding h : ˜ S → M with trivialized normal bundleextending the canonical diffeomorphism ˜ σ (cid:48) → ˜ σ and the trivialization of itsnormal bundle respectively such that h − and h + are right inverses to f .Proof. We may assume that f | Σ f is a general position immersion. Let S j denote the submanifold in S of points of f (Σ f ) of multiplicity j and S is the complement to ∪ S i in S . Then S = ∪ S j . Suppose that h − , h + andtrivializations have been constructed over a neighborhood of S j for all j > k .Let D be an open tubular neighborhood of Σ f in M . Then over B = S k a component of S a component of S a component of S Figure 10.
De composition of S .the map b given by f | M \ D is a fiber bundle with fiber F g, k for some g . By Example 7.1, there is a bordism b : E → B of b to b : E → B such that h − , h + and trivializations extend over B . The bordism b canbe essentially uniquely thickened to a bordism b : E → B = B × D k ofthe restriction of f over a disc neighborhood of B so that b is a brokensubmersion with breaking fold values (cid:116) B × D k − i where D k − i ranges overall k coordinate hyperdiscs in D k . Let N be the union of N × I and B inwhich the top submanifold ( B × D k ) × { } is identified with B × D k ⊂ B .Let M be a similar union of M × I and E . Then after smoothing cornerswe obtain a bordism f = f × id I ∪ b of f to f such that h − , h + andtrivializations extend over a neighborhood of S k ( f ). Thus, by induction,we get a desired extension. (cid:3) Remark . The above construction works in the case of N = S as well.Indeed, choose S to be the interval in N over which the fibers of f are ofmaximal Euler characteristic. Then the above bordism eliminates the twofolds in f − ( ∂S ). Continuing by induction we end up with a submersion.Note that here the bordism of f is actually a concordance. HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 13
Figure 11.
Trading singularities.
Figure 12.
The component σ can be “traded” for a newcomponent of breaking folds parallel to ∂N . Proposition 7.6.
Let f be a broken submersion M → N to a compactsimply connected manifold N . Suppose that over ∂N the map f is a fiberbundle with fiber F g of genus g (cid:29) dim N . Then f | ∂N is zero bordant inthe class of fiber bundles.Proof. In view of Propositions 5.3 and 6.1, we may assume that the fiber of f over each regular point is a connected surface of genus (cid:29) dim N and that f has no folds of index 0. Let σ be a component of folds, and S a closeddomain bounded by f ( σ ). Assume that the coorientation of ∂S is outwarddirecting; otherwise σ can be eliminated as above. We may assume that aneighborhood of ∂N is identified with ∂N × [0 ,
2) and over U = ∂N × [0 , f is the trivial concordance of f | ∂N . Modify f over U so that it is a concordance that first stabilizes the fibers and thendestabilizes them back, see Example 5.1. Then f has two new componentsof breaking folds. One of these two components can be eliminated with σ by the concordance as above. Thus the component σ can be “traded” fora new component of breaking folds parallel to ∂N . Consequently, we mayassume that f only has breaking folds parallel to ∂N .In other words, the map f over a collar neighborhood of ∂N is a concor-dance that stabilizes the fibers, and over the complement to the collar neigh-borhood of ∂N it is a fiber bundle. It remains to apply Example 7.2. (cid:3) The weak b-principle
A collection C of smooth maps f : M → N with fixed dim M − dim N = d is said to be a class of maps of dimension d if the induced map h ∗ g in thepullback diagram M (cid:48) −−−−→ M h ∗ g (cid:121) g (cid:121) N (cid:48) h −−−−→ N. is in C for every map g ∈ C and every map h transverse to g . Example 8.1. If g : M → N is a submersion, then for every smooth map g : N (cid:48) → N , the induced map h ∗ g is a submersion as well. If g is animmersion, then the induced map h ∗ g is an immersion as well provided that h is transverse to g , i.e., provided that for each x ∈ N , x (cid:48) ∈ N (cid:48) and y ∈ M such that h ( x (cid:48) ) = x = g ( y ), we haveIm( d x (cid:48) h ) ⊕ Im( d y g ) (cid:39) T x N. Thus, both submersions and immersions of dimension d form classes of maps.More generally, solutions to any open stable differential relation R form aclass of maps [14]. The transversality condition is clearly important here:if a smooth map h is not transverse to a smooth map g , then the pullbackspace M (cid:48) may not admit a manifold structure.An appropriate quotient space of all proper maps in a collection C is calledthe moduli space for C . Namely, recall that the opening of a subset X ofa manifold V is an arbitrarily small but non-specified open neighborhoodOp( X ) of X in V . Consider the affine subspace { x + · · · + x m +1 = 1 } ⊂ R m +1 . It contains the standard simplex ∆ m bounded by all additional conditions0 ≤ x i ≤
1. Let ∆ ne denote the opening of ∆ m in the considered affinesubspace. Then every morphism δ in the simplicial category extends linearlyto a map ˜ δ : ∆ me → ∆ ne . Let X m denote the subset of C of proper maps to∆ me transverse to all extended face maps. Then X • is a simplicial set withstructure maps X ( δ ) given by the pullbacks f (cid:55)→ ˜ δ ∗ f .The (simplicial model of the) moduli space M for C is the semi-simplicialgeometric realization of X • . We say that C satisfies the sheaf property if f : M → N belongs to C whenever each f | f − U i is in C for a covering { U i } of N . If f satisfies the sheaf property, then the sets Ω ∗ M and [ N, M ] areisomorphic to the sets of bordism classes and concordance classes of propermaps in C to N respectively.We say that a class C is monoidal if the map of the empty set to a pointis a map in C and the class C is closed with respect to taking disjoint unionsof maps, i.e., if f : M → N and f : M → N are maps in C , then f (cid:116) f : M (cid:116) M −→ N HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 15
Figure 13.
A map ( f, α ) in the collection h C .is also a map in C . For a monoidal class C the space M is an H-space witha coherent operation (i.e., the first term of a Γ-space). We will recall theconstruction of its classifying space M and an approximation of M by aspace h M of a relatively simpler homotopy type, for details see [14], [15].Let C be the derived collection (not a class) of proper maps ( f, α ) : V → N × R with f ∈ C such that every regular fiber of ( f, α ) is null-cobordant; andlet C ⊂ h C be a subcollection of pairs with α ◦ f − ( x ) (cid:54) = R for all x ∈ N .The spaces M and h M are the geometric realizations of simplicial sets ofmaps ( f, α ) to ∆ me × R such that f is transverse to all extended face mapsand ( f, α ) is in C and h C respectively. Definition 8.2.
The weak b-principle for C is said to hold true if the inclu-sion M → h M is a homotopy equivalence. Theorem 8.3 (Sadykov, [15]) . Let C be a monoidal class of maps satisfyingthe sheaf property. Suppose that every breaking concordance of every mapin h C is itself in h C . Then the weak b-principle for C holds true. Under the assumptions of Theorem 8.3, if M is path connected, then itis homotopy equivalent to its group completion Ω M . Furthermore, in viewof Theorem 8.3, we can identify M with Ω h M .9. Colored broken submersions
A map f : M → N may not be a broken submersion even if its restrictionto every subset f − ( U i ) for an open covering { U i } of N is a broken submer-sion. In other words, broken submersions do not satisfy the sheaf property .We will use colored broken submersions that satisfy the sheaf property.Let I denote the category of finite sets n = { , ..., n } for n ≥ m (cid:116) n of objects in I . An m -coloring ona broken submersion f is a map C f from the set of path components ofbreaking folds of f to the set m such that the restriction of f to breakingcomponents of any fixed color is an embedding; here we allow m to be anyelement in I or the set ∞ of positive integers. The moduli space of m -colored broken submersions is denoted by M m . Recall that an I -space is a functor I →
Top . We are interested in the I -space m (cid:55)→ M m ; its hocolimis denoted by M b , see [18]. Theorem 9.1.
The set of oriented bordism classes of broken submersionsof dimension over closed oriented manifolds of dimension n is naturallyisomorphic to Ω n ( M b ) .Proof. Given a broken submersion f over an oriented closed manifold, achoice of a coloring on its folds determines a class τ ( f ) in Ω ∗ ( M b ). Wemay choose a coloring so that different breaking components are colored bydifferent colors. Then, since every isomorphism m → m is a morphism in I , the class τ ( f ) does not depend on the choice of the coloring. If f isbordant to a broken submersion g , then we may assume that the images ofthe classifying maps of f and g are in M m for a sufficiently big palette m and therefore τ ( f ) = τ ( g ). Conversely, every map τ : N → M b representinga bordism class in Ω ∗ ( M b ) is linearly homotopic to a map with image in M m for some sufficiently big palette m , and therefore every map τ determines acolored broken submersion. (cid:3) The same argument shows that the canonical map of the telescope M ∞ =colim M m to M b and the canonical map M b → M ∞ are homotopy equiv-alences. In particular, homotopy classes [ N, M b ] are in bijective correspon-dence with concordance classes of ∞ -colored broken maps to N . Similarly,the homotopy colimit of the I -space m (cid:55)→ M m is denoted by M b andcolim M m (cid:39) M b .A general argument on I -spaces shows that M b is an infinite loop space,see [18]. Alternatively, the Galatius-Madsen-Tillmann-Weiss argument in[14] shows that M b is an infinite loop space, and its classifying space is M b .The H-space operation on M b is defined by M m × M n −→ M m (cid:116) n , ∆ f × ∆ g (cid:55)→ ∆ f (cid:116) g , where ∆ h is the simplex in the moduli space corresponding to a map h . Wechoose the unit point to be the vertex in M ∅ ⊂ M b corresponding to themap ∅ → ∆ e .Since M b is path connected, we have M b (cid:39) Ω M b . Furthermore, byTheorem 8.3 the weak b-principle for colored broken submersions holds true.Consequently, M b (cid:39) Ω h M b .10. Proof of the Mumford conjecture
Proof of Theorem 1.1.
Let h M (cid:39) Ω ∞ MTSO(2) be the moduli space for ori-ented stable formal submersions of dimension 2. We need to show that themap BDiff F g → h M induces an isomorphism of homology groups in dimen-sions (cid:28) g . Recall that h M is the geometric realization of the simplicial setwhose simplicies are given by pairs of proper maps ( f, α ) to ∆ ne × R suchthat f is a submersion of dimension 2, see [14]. The simplicies of a bigger HE WEAK B-PRINCIPLE: MUMFORD CONJECTURE 17 simplicial complex h M b correspond to proper maps ( f, α ) to ∆ ne × R suchthat f is a broken submersion of dimension 2 whose components of folds arelabeled. Hence, there is an inclusion h M → h M b , which defines a map ofthe loop space h M (cid:39) Ω h M to the loop space Ω h M b (cid:39) M b . Hence, weget a sequence of maps η : BDiff F g −→ h M −→ M b . Since M b is an H-space, its fundamental group is abelian and thereforeequals [ S , M b ]. On the other hand, every broken submersion over S isconcordant to a fiber bundle with fiber F g , see Remark 7.5. Hence, thefundamental group of M b is the image of the perfect group π (BDiff F g )provided that g ≥
3. Consequently, the space M b is simply connected. Inparticular, every bordism class of M b is represented by a map of a simplyconnected manifold N . By Proposition 7.3, every broken submersion overa closed simply connected manifold N is bordant to a fiber bundle withfiber F g . Thus, η induces an epimorphism in integral homology groups indimensions n (cid:28) g .Let us show that η ∗ is injective in dimensions n (cid:28) g , i.e., given a brokensubmersion f over N which restricts over ∂N to a fiber bundle with fiber F g of genus g (cid:29) dim N , there is a fiber bundle f over N that restricts over ∂N = ∂N to f | ∂N . Again, we may assume that N is simply connected.Thus, the statement follows from Proposition 7.6. This implies that η ∗ is anisomorphism in integral homology groups in dimensions (cid:28) g . Consequently,the b-principle map BDiff F g → h M induces an injective homomorphism inhomology groups in a stable range. On the other hand, by the Miller-Moritatheorem, the induced homomorphism in rational homology groups is alsosurjective in a stable range [12]. This implies the Mumford conjecture. (cid:3) Proof of Theorem 1.3.
We may turn the map η : BDiff F g → M b defined inthe proof of Theorem 1.1 into a cofibration. Then the pair ( M b , BDiff F g )classifies bordism classes( f, ∂f ) : ( M, ∂M ) −→ ( N, ∂N )such that f is a smooth broken submersion over N that restricts over theboundary ∂N to a fiber bundle ∂f with fiber F g , dim N (cid:28) g . It remains toobserve that Ω ∗ ( M b , BDiff F g ) = 0 for ∗ (cid:28) g since η ∗ is an isomorphism ina stable range. (cid:3) References [1] Y. Ando, Cobordisms of maps with singularities of a given class, Alg. Geom. Topol., 8 (2008),1989–2029.[2] M. Audin, Cobordismes d’immersions lagrangiennes et legendriennes, th`ese d’´etat, Orsay,1986, Travaux en cours, Hermann, Paris, 1987.[3] M. Barratt, P. Priddy, On the homology of non-connected monoids and their associatedgroups, Commentarii Math. Helvetici 47 (1972), 1–14.[4] Y. Eliashberg, S. Galatius, N. Mishachev, Madsen-Weiss for geometrically minded topolo-gists, Geometry and Topology 15 (2011) 411–472.8 RUSTAM SADYKOV[5] Y. Eliashberg, Cobordisme des solutions de relations diff´erentielles, South Rhone seminar ongeometry. I, Lyon, 1983, Trav. Cours, 1984, 17–31.[6] Y. Eliashberg, N. Mishachev, Introduction to the h-principle, Grad. Stud. Math, v48, AMS,Providence, Rhode Island, 2002.[7] S. Galatius, I. Madsen, U. Tillmann, M. Weiss, The homotopy type of the cobordism category,Acta Math. 202 (2009), 195–239.[8] S. Galatius, O. Randal-Williams, Monoids of moduli spaces of manifolds, Geometry andTopology 14 (2010) 1243–1302.[9] M. Gromov, Partial differential relations, Springer-Verlag, Berlin, Heidelberg, 1986.[10] A. Hatcher, A short exposition of the Madsen-Weiss theorem, preprint.[11] I. Madsen, M. S. Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture,Ann. of Math. (2) 165 (2007), 843–941.[12] Sh. Morita, Geometry of characteristic classes, Translations of Math. Monographs 199, AMS,2001.[13] R. Rim´anyi, A. Sz˝ucs, Pontrjagin-Thom type construction for maps with singularities, Topol-ogy, 37 (1998), 1177–1191.[14] R. Sadykov, The bordism version of the h-principle, preprint.[15] R. Sadykov, The weak b-principle, Contemporary Math. 621 (2014), 101–112.[16] R. Sadykov, Bordism groups of special generic mappings, Proc. Amer. Math. Soc., 133 (2005),931–936.[17] R. Sadykov, Bordism groups of solutions to differential relations, Alg. Geom. Topol., 9 (2009),2311–2349.[18] Ch. Schlichtkrull, Units of the ring spectra and thire traces in algebraic K-theory, Geom.Topol. 8 (2004), 645–673.[19] A. Sz˝ucs, Cobordism of singular maps, Geom. and Topol., 12 (2008), 2379–2452.[20] R. Wells, Cobordism groups of immersions, Topology, 5 (1966), 281–294.
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