The moduli space of generalized Morse functions
aa r X i v : . [ m a t h . A T ] A p r THE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS
BORIS BOTVINNIK AND IB MADSEN
Abstract.
We study the moduli and determine a homotopy type of the space of all gener-alized Morse functions on d -manifolds for given d . This moduli space is closely connectedto the moduli space of all Morse functions studied in [11] and the classifying space of thecorresponding cobordism category. Introduction
Given a smooth compact manifold M d and a fixed smooth function ϕ : M d → R , let G ( M d , ϕ )denote the space of generalized Morse functions f : M → R which agrees with ϕ in aneighbourhood of the boundary ∂M . This space (in the Whitney topology) satisfies an h -principle in the sense of Gromov, [5]. Here precisely we define h G ( M d , ϕ ) to be the space ofsections of the bundle J ( M ) of generalized Morse 3-jets that agrees with j ϕ near ∂M .Taking the 3-jet of a generalized Morse function defines a map(1) j : G ( M d , ϕ ) −→ h G ( M d , ϕ ) . This map was first considered by Igusa in [6]. He proved that the map j in (1) is d -connectedand in [7] he calculated the “ d -homotopy type” of h G ( M d , ϕ ) by exhibiting a d -connectedmap(2) h G ( M d , ϕ ) −→ Ω ∞ S ∞ ( BO + ∧ M d ) , thus determining the d -homotopy type of the space G ( M d , ϕ ).Eliashberg and Mishachev [2, 3] and Vassiliev [13] showed that the map in (1) is actually ahomotopy equivalence rather than just being d -connected. This is the starting point for thispaper.We study the moduli space of all generalized Morse functions on d -manifolds, i.e. the spaceof G ( M d , ϕ ) as ( M d , ϕ ) varies. There can be several candidates for such a moduli space.The one we present below is closely connected to the “moduli space” of all Morse functionsconsidered in Section 4 of [11]. Indeed, the present note can be viewed as an addition to [11].In Section 2 below we give the precise definition of our moduli space, and in Section 3 wedetermine its homotopy type, following the argument from [11]. Boris Botvinnik is partially supported by SFB-748, M¨unster, Germany.Ib Madsen is partially supported by ERC advanced grant 228082-TMSS. Definitions and results
The moduli space.
Let J ( R d ) be the space of 3-jets of smooth functions on R d , p ( x ) = c + ℓ ( x ) + q ( x ) + r ( x ) , where c is a constant, ℓ ( x ) is linear, q ( x ) quadratic and r ( x ) cubic, ℓ ( x ) = X i a i x i , q ( x ) = X ij a ij x i x j , r ( x ) = X ijk a ijk x i x j x k , with the coefficients a ij , a ijk symmetric in the indices.Let J ( R d ) ⊂ J ( R d ) be the subspace of p ∈ J ( R d ) such that one the following holds:(i) 0 ∈ R d is not a critical point of p ( ℓ = 0);(ii) 0 ∈ R d is a non-degenerate critical point of p ( ℓ = 0 and q is non-degenerate);(iii) 0 ∈ R d is a birth-death singularity of p ( q : R d → Hom R ( R d , R ) has 1-dimensionalkernal on which r ( x ) is non-trivial).The space J ( R d ) is invariant under the O ( d )-action on the space J ( R d ). Given a smoothmanifold M d with a metric, let P ( M d ) → M be the principal O ( d )-bundle of orthogonalframes in the tangent bundle T M d . Then J ( T M d ) = P ( M d ) × O ( d ) J ( R d )is a smooth fiber bundle on M , a subbundle of J ( T M d ) = P ( M d ) × O ( d ) J ( R d ) . Remark 2.1.
Up to change of coordinates, a birth-death singularity is of the form p ( x ) = x − i +1 X j =2 x j + d X k = i +2 x k . The integer i is the Morse index of the quadratic form q ( x ). In general, a quadratic form q : R d → R induces a canonical decomposition R d = V − ( q ) ⊕ V ( q ) ⊕ V + ( q )into negative eigenspace, the zero eigenspace and the positive eigenspace. In the case ofgeneralized Morse jets, dim V ( q ) is either 0 or 1, and in the latter case the cubic term r ( x )restricts non-trivially to V ( q ). The dimension of V − ( q ) is the index of the gmf-jet. ✸ For a smooth manifold M d , we have the 3-jet bundle J ( M, R ) → M whose fiber J ( M, R ) x is the germ of 3-jets ( M, x ) → R , and the associated subbundle J ( M, R ) → M . A choiceof exponential function induces a fiber bundle isomorphism(3) J ( M, R ) ∼ = J ( T M ) , HE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS 3 and f : M → R is a generalized Morse function precisely if j ( f ) ∈ Γ( J ( M, R )), where weuse Γ( E ) to denote the space of smooth sections of a vector bundle E . Definition 2.1.
Let X be a k -dimensional manifold without boundary. Let J d ( X ) be theset of 4-tuples ( E, π, f, j ) of a ( k + d )-manifold E with maps( π, f, j ) : E −→ X × R × R d − ∞ subject to the conditions(i) ( π, f ) : E −→ X × R is a proper map;(ii) ( f, j ) : E −→ R × R d − ∞ is an embedding;(iii) π : E −→ X is a submersion;(iv) for any x ∈ X , the restriction f x = f | E x : E x −→ R to each fiber E x = π − ( x ) is ageneralized Morse function.In (ii) above R d − ∞ is the union or colimit of R d − N as N → ∞ and (ii) means that ( f, j )embeds E into R × R d − N for sufficiently large N . The definition above is the obviousanalogue of Definition 2.7 of [11].A smooth map φ : Y −→ X induces a pull-back φ ∗ : J d ( X ) −→ J d ( Y ) , φ ∗ : ( E, π, f, j ) ( φ ∗ E, φ ∗ π, φ ∗ f, φ ∗ j ) where φ ∗ E = { ( y, z ) ∈ Y × R d + ∞ | ( φ ( y ) , z ) ∈ E ⊂ X × R d + ∞ } and the maps φ ∗ π , φ ∗ f and φ ∗ j are given by corresponding projections from φ ∗ E ⊂ Y × R × R d − ∞ on the factors Y , R and R d − ∞ , respectively. In particular, we have ( ψ ◦ φ ) ∗ = φ ∗ ◦ ψ ∗ (ratherthan just being naturally equivalent), so the correspondence X
7→ J d ( X ) is a set-valued sheaf J d on the category X of smooth manifolds and smooth maps.A set-valued sheaf on X gives rise to a simplicial set N • J d with N k J d = J d (∆ ke ) , ∆ ke = (cid:8) ( x , . . . , x k ) ∈ R k +1 | P x i = 1 (cid:9) . The geometric realization of N • J d will be denoted by |J d | , and we make the following defi-nition. Definition 2.2.
The moduli space of generalized Morse functions of d variables is the loopspace Ω |J d | . Remark 2.2.
If in Definition 2.1 we drop the assumption that f : E −→ R is a generalizedMorse function, then J d reduces to the sheaf D d = D d ( − , ∞ ) of [4, Definition 3.3] associatedto the space of embedded d -manifolds, and Ω | D d | ∼ = Ω ∞ M T ( d ) by Theorem 3.4 of [4]. ✸ BORIS BOTVINNIK AND IB MADSEN
Associated with the set valued sheaf J d ( X ), X ∈ X , we have a sheaf J A d ( X ) of partiallyordered sets, i.e. a category valued sheaf, cf. [11, Section 4.2]. For connected X , an object of J A d ( X ) consists of an element ( E, π, f, j ) ∈ J d ( X ) together with an interval A = [ a , a ] ⊂ R subject to the condition that f : E −→ R be fiberwise transverse to ∂A (i.e. { a , a } areregular values for each f x : E x −→ R , x ∈ X ). The partial ordering is given by( E, π, f, j ; A ) ≤ ( E ′ , π ′ , f ′ , j ′ ; A ′ ) if( E, π, f, j ) = ( E ′ , π ′ , f ′ , j ′ ) and A ⊂ A ′ . If X is not connected, J A d ( X ) is the product of J A d ( X j ) over the connected components X j .We notice that each element ( E, π, j, f ; A ) restricts to a family of generalized Morse functionson a compact manifolds ( π, f, j ) : f − ( A ) ֒ → X × [ a , a ] × R d − ∞ , where A = [ a , a ]. On the other hand, given such a family( π, f, j ) : E ( A ) ֒ → X × [ a , a ] × R d − ∞ , we can extend it to an element ( ˆ E ( A ) , ˆ π, ˆ f , ˆ j ) by adding long collars:ˆ E ( A ) = ( −∞ , a ] × f − ( a ) ∪ E ( A ) ∪ [ a , ∞ ) × f − ( a ) . If ( E ( A ) π, f, j ) is a restriction of ( E, π, f, j ) ∈ J d ( X ), then ( ˆ E ( A ) , ˆ π, ˆ f , ˆ j ; A ) is concordantto ( E, π, f, j ) by [11, Lemma 2.19].The forgetful map J A d ( X ) −→ J d ( X ) is a map of category valued sheaves when we give J d ( X ) the trivial category structure (with only identity morphisms). It induces a map |J A d | −→ |J d | of topological categories, and hence a map of their classifying spaces: B |J A d | −→ B |J d | = |J d | , where B |J | = | N • J | . Theorem 2.1.
The map B |J A d | −→ |J d | is a weak homotopy equivalence. Proof . This follows from [11, Theorem 4.2], which identifies | B J A d | with | β J A d | , where β J A d is a set-valued sheaf of [11, Definition 4.1], together with the analogue of [11, Proposition 4.10]:the map | β J A d | −→ |J A d | is a weak homotopy equivalence. Indeed, the proof of Proposition 4.10, which treats the casewhere f : E −→ R is a fiberwise Morse function (in a neighbourhood of f − (0)) carries overword by word to the situation of generalized Morse functions. (cid:3) HE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS 5
The h -principle. For a submersion π : E −→ X , T π E denotes the tangent bundlealong the fibers. We can form the bundle J ( T π E ) −→ E of gmf-jets. Sections of this bundle will be denoted by ˆ f , ˆ g, . . . etc. For given z ∈ E , therestriction ˆ f ( z ) : T ( E π ( z ) ) −→ R is a gmf-jet; its constant term will be denoted f ( z ). Definition 2.3.
For a smooth manifold X , let h J d ( X ) consists of maps( π, f, j ) : E −→ X × R × R d − ∞ satisfying (i), (ii), and (iii) of Definition 2.1 together with a jet ˆ f ∈ Γ( J ( T π E )) havingconstant term f . ✷ A metric on T π E and an associated exponential map induces an isomorphism(4) J π ( E, R ) ∼ = −→ J ( T π E )that sends gmf-jets to gmf-jets. Here J π ( E, R ) −→ E is the fiberwise 3-jet bundle. Differen-tiation in the fiber direction only, defines a map j π : C ∞ ( E, R ) −→ J π ( E, R )which sends fiberwise generalized Morse functions into gmf-jets. This induces a map of sheaves j π : J d ( X ) −→ h J d ( X ) , and hence a map of their nerves j π : |J d | −→ | h J d | . Using the sheaves J A d and the associated h J A d , together with the h -principle of [3, 13], theargument of [11, Proposition 4.17] proves that the induced map | β J A d | −→ | βh J A d | is a weak homotopy equivalence. Finally, since the forgetful maps | β J A d | −→ |J A d | , | βh J A d | −→ | h J A d | are weak homotopy equivalences by [11, Proposition 4.10], we get Theorem 2.2.
The map j π : |J d | −→ | h J d | is weak homotopy equivalence. ✷ BORIS BOTVINNIK AND IB MADSEN Homotopy type of the moduli space
The space | h J d | . For any set-valued sheaf F : X −→ S ets , let F [ X ] denote theset of concordance classes: s , s ∈ F ( X ) are concordant ( s ∼ s ) if there exists an element s ∈ F ( X × R ) such that pr ∗ X ( s ), pr X : X × R −→ X , agrees with s on an open neighborhoodof X × ( −∞ ,
0] and pr ∗ X ( s ) agrees with s on an open neighborhood of X × [1 , ∞ ). Therelation to the space |F | is given by(5) [ X, |F | ] ∼ = F [ X ] . Let G ( d, n ) denote the Grassmannian of d -planes in R d + n , and G gmf ( d, n ) the space of pairs( V, f ) with V ∈ G ( d, n ) and f : V → R a generalized Morse function with f (0) = 0. Thespace U ⊥ d,n = { ( v, V ) ∈ R d + n × G ( d, n ) | v ⊥ V } is an n -dimensional vector bundle on G ( d, n ). Let V ⊥ d,n be its pull-back along the forgetfulmap G gmf ( d, n ) −→ G ( d, n ): V ⊥ d,n U ⊥ d,n G gmf ( d, n ) G ( d, n ) ✲❄ ❄✲ Similarly, we have canonical d -dimensional vector bundles U d,n and V d,n on G ( d, n ) and G gmf ( d, n ) respectively. The Thom spaces of the bundles U ⊥ d,n and V ⊥ d,n give rise to spectra M T ( d ) and M T gmf ( d ) which in degrees ( d + n ) are(6) M T ( d ) d + n = Th ( U ⊥ d,n ) , M T gmf ( d ) d + n = Th ( V ⊥ d,n ) . The infinite loop space of the spectrum
M T gmf ( d ) is defined to be(7) Ω ∞ M T gmf ( d ) = colim n Ω d + n Th ( V ⊥ d,n ) Theorem 3.1.
There is a weak homotopy equivalence (8) Ω | h J d | ≃ Ω ∞ M T gmf ( d ) . Proof . This is completely similar to the proof of [11, Theorem 3.5] for the case of Morsefunctions using only transversality and the submersion theorem, [12]. (cid:3)
We have left to examine the right-hand side of the equivalence (8). The results are similar inspirit to ones in [11, Section 3.1].Let Σ gmf ( d, n ) ⊂ G gmf ( d, n ) be the singular set of pairs ( V, f ) with f : V −→ R havingvanishing linear term, i.e. Df (0) = 0. Consider the nonsingular set G gmf ( d, n ) \ Σ gmf ( d, n )given as set of pairs ( V, f ) ∈ G gmf ( d, n ) with f = ℓ + q + r and ℓ = 0. We notice that thespace G gmf ( d, n ) \ Σ gmf ( d, n ) retracts to the spaceˆ G gmf ( d, n ) = { ( V, f ) ∈ G gmf ( d, n ) | f = ℓ + q + r, with | ℓ | = 1 , q = 0 , r = 0 } , HE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS 7 where | ℓ | is a norm of the linear part ℓ : V −→ R . Lemma 3.2.
There is a homeomorphism ˆ G gmf ( d, n ) ∼ = O ( d + n ) / ( O ( d − × O ( n )) . Proof . For a pair (
V, f ) ∈ ˆ G gmf ( d, n ) we have V ∈ G ( d, n ) and f = ℓ with | ℓ | = 1. Wemay think of ℓ as a linear projection on the first coordinate, which is the same as to say thatthe space V contains a subspace(9) R × { } × { } ⊂ R × R d − × R n with ℓ being a projection on it. This identifies ˆ G gmf ( d, n ) with the homogeneous space O ( d + n ) / ( O ( d − × O ( n )). (cid:3) Since G ( d − , n ) = O ( d − n ) / ( O ( d − × O ( n )), we observe that the map i d,n : G ( d − , n ) −→ ˆ G gmf ( d, n )is ( d + n − i ∗ d,n ( V ⊥ d,n | ˆ G gmf ( d,n ) ) ∼ = U ⊥ d − ,n .On the other hand, Σ gmf ( d, n ) ⊂ G gmf ( d, n ) has normal bundle V ∗ d,n ∼ = V d,n and the inclusion( D ( V d,n ) , S ( V d,n )) −→ ( G gmf ( d, n ) , G gmf ( d, n ) \ Σ gmf ( d, n ))is an excision map. This leads to the cofibration(10) Th ( j ∗ V ⊥ d,n ) −→ Th ( V ⊥ d,n ) −→ Th (( V ⊥ d,n ⊕ V d,n ) | Σ gmf ( d,n ) ) , where j is the inclusion j : G gmf ( d, n ) \ Σ gmf ( d, n ) −→ G gmf ( d, n ) . By the above, there is (2 n + d − i d,n : Th ( U ⊥ d − ,n ) −→ Th ( V ⊥ d,n ) . With the notation of (6), we get from (10) a cofibration of spectra(11) Σ − M T ( d − −→ M T gmf ( d ) −→ Σ ∞ (Σ gmf ( d, ∞ ) + )and the corresponding homotopy fibration sequence of infinite loop spacesΩ ∞ Σ − M T ( d − −→ Ω ∞ M T gmf ( d ) −→ Ω ∞ Σ ∞ (Σ gmf ( d, ∞ ) + ) Remark 3.1.
The main theorem of [4] asserts a homotopy equivalenceΩ ∞ M T ( d ) ≃ Ω B C ob d , where C ob d is the cobordism category of embedded manifolds: the objects are ( M d − , a ) of aclosed ( d − { a } × R ∞ + d − and the morphisms are embedded cobordisms W d ⊂ [ a , a ] × R ∞ + d − transversal at { a i } × R ∞ + d − . In particular, we have weak homotopyequivalence(12) Ω ∞ Σ − M T ( d − ≃ Ω B C ob d − . BORIS BOTVINNIK AND IB MADSEN
The singularity space.
In [7], Igusa analyzed the singularity space Σ gmf ( d ) ⊂ J ( R d )by decomposing it with respect to the Morse index. The result, stated in [7, Proposition 3.4],is as follows. Consider the homogeneous spaces X ( i ) = O ( d ) . O ( i ) × O (1) × O ( d − i − ,X ( i ) = O ( d ) . O ( i ) × O ( d − i ) . and note that there are quotient maps f i : X ( i ) → X ( i ) , g i : X ( i ) → X ( i + 1) , upon embedding O ( i ) × O (1) × O ( d − i −
1) in O ( i ) × O ( d − i ) and in O ( i + 1) × O ( d − i − D ( d ) = X (0) X (1) · · · X ( d − X (0) X (1) X (2) X ( d − X ( d ) ❅❅❅❘ g (cid:0)(cid:0)(cid:0)✠ f ❅❅❅❘ g (cid:0)(cid:0)(cid:0)✠ f ❅❅❅❘ g d − (cid:0)(cid:0)(cid:0)✠ f ❅❅❅❘ g d − (cid:0)(cid:0)(cid:0)✠ f d − and [7, Proposition 3.4] states that the homotopy colimit of the diagram D ( d ) is homotopyequivalent to Σ gmf ( d )(13) Σ gmf ( d ) ≃ hocolim D ( d ) . It is easy to see that there are homeomorphisms G gmf ( d, n ) = (cid:16) O ( d + n ) . O ( n ) (cid:17) × O ( d ) J ( R d ) , Σ gmf ( d, n ) = (cid:16) O ( d + n ) . O ( n ) (cid:17) × O ( d ) Σ gmf ( d ) , and (13) implies that Σ gmf ( d, n ) is homotopy equivalent to the homotopy colimit of the dia-gram D ( d, n ) = (cid:16) O ( d + n ) . O ( n ) (cid:17) × O ( d ) D ( d ) . For n → ∞ , the Stiefel manifold O ( n + d ) /O ( n ) becomes contractible, and D ( d, ∞ ) is thediagram(14) Y (0) Y (1) · · · Y ( d − Y (0) Y (1) Y (2) Y ( d − Y ( d ) ❅❅❅❘ ¯ g (cid:0)(cid:0)(cid:0)✠ ¯ f ❅❅❅❘ ¯ g (cid:0)(cid:0)(cid:0)✠ f ❅❅❅❘ ¯ g d − (cid:0)(cid:0)(cid:0)✠ ¯ f ❅❅❅❘ ¯ g d − (cid:0)(cid:0)(cid:0)✠ ¯ f d − HE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS 9 with Y ( i ) = BO ( i ) × BO (1) × BO ( d − i − ,Y ( i ) = BO ( i ) × BO ( d − i ) , and ¯ f i and ¯ g i the obvious maps. So Σ gmf ( d, ∞ ) is the homotopy colimit of (14).We want to compare this to the singular set Σ mf ( d, ∞ ) which appears when one considersthe moduli space of Morse functions rather than generalized Morse functions was calculatedin [11, Lemma 3.1]:Σ mf ( d, n ) ∼ = d Y i =0 h(cid:16) O ( d + n ) . O ( n ) (cid:17) × O ( d ) (cid:16) O ( d ) . O ( i ) × O ( d − i ) (cid:17)i so that Σ mf ( d, ∞ ) ∼ = d Y i =0 BO ( i ) × BO ( d − i ) = d Y i =0 Y ( i ) . The cofiber of the map Σ mf ( d, ∞ ) −→ Σ gmf ( d, ∞ ) is by (14) equal to the homotopy colimitof the diagram: Y (0) Y (1) · · · Y ( d − ∗ ∗ ∗ ∗ ∗ ❙❙❙❙✇✓✓✓✓✴ ❙❙❙❙✇✓✓✓✓✴ ❙❙❙❙✇✓✓✓✓✴ ❙❙❙❙✇✓✓✓✓✴ But this homotopy colimit is easy calculated to be d − _ i =0 S ∧ Y ( i ) + ≃ d − _ i =0 S ∧ ( BO ( i ) × BO ( d − i − + . We get a cofibration of suspension spectra:Σ ∞ (Σ mf ( d, ∞ ) + ) −→ Σ ∞ (Σ gmf ( d, ∞ ) + ) −→ d − _ i =0 Σ ∞ ( S ∧ ( BO ( i ) × BO ( d − i − + )) . Taking the associated infinite loop spaces we get
Proposition 3.3.
There is a homotopy fibration: d − Y i =0 Ω ∞ Σ ∞ ( BO ( i ) × BO ( d − i − + ) → d Y i =0 Ω ∞ Σ ∞ (Σ mf ( d, ∞ ) + ) → Ω ∞ Σ ∞ (Σ gmf ( d, ∞ ) + ) . ✷ The constant map D ( d ) → ∗ into the constant diagram induces the map D ( d, n ) → (cid:16) O ( d + n ) . O ( n ) (cid:17) × O ( d ) ∗ , where the target space is homotopy equivalent to G ( d, n ). For n → ∞ , this induces the fiberbundle(15) p : Σ gmf ( d, ∞ ) → BO ( d )with the fiber Σ gmf ( d ). We obtain the commutative diagram of cofibrations:Σ − M T ( d − M T gmf ( d ) Σ ∞ (Σ gmf ( d, ∞ ) + )Σ − M T ( d − M T ( d ) Σ ∞ ( BO ( d ) + ) ✲❄ Id ✲❄ F ❄ Σ ∞ p ✲ ✲ and a corresponding diagram of homotopy fibrations:(16) Ω ∞ Σ − M T ( d −
1) Ω ∞ M T gmf ( d ) Ω ∞ Σ ∞ (Σ gmf ( d, ∞ ) + )Ω ∞ Σ − M T ( d −
1) Ω ∞ M T ( d ) Ω ∞ Σ ∞ ( BO ( d ) + ) ✲❄ Id ✲❄ Ω ∞ F ❄ Ω ∞ Σ ∞ p ✲ ✲ Since Σ ∞ (Σ gmf ( d, ∞ ) + ) and Σ ∞ ( BO ( d ) + ) are ( − F : M T gmf ( d ) → M T ( d ) induces isomorphism π − i M T gmf ( d ) ∼ = π − i M T ( d ) , i ≥ . We consider the forgetful map θ gmf : G gmf ( d, ∞ ) −→ G ( d, ∞ ) as a structure on d -dimensionalbundles . Then we denote by C ob gmf d the category C ob θ gmf d (see (5.3) and (5.4) of [4]) of manifolds(objects) and cobordisms (morphisms) equipped with a tangential θ gmf -structure. Then themain theorem of [4] gives the following result: Corollary 3.4.
There is weak homotopy equivalence B C ob gmf d ∼ = Ω ∞− M T gmf ( d ) , and the forgetting map B C ob gmf d → B C ob d induces isomorphism Ω gmf d = π B C ob gmf d ≃ π B C ob d = Ω d , where Ω gmf d and Ω d are corresponding cobordism groups. Remarks on the moduli space of Morse functions.
The paper [11] studied themoduli space of fiberwise Morse functions. The fibers are the space of functions which locallyhas 2-jets of the form f : R d → R , f = f (0) + ℓ ( x ) + q ( x ) subject to the conditions:(i) f (0) = 0 or(ii) f (0) = 0 and ℓ ( x ) = 0 or(iii) f (0) = 0, ℓ ( x ) = 0 and q ( x ) is non-singular quadratic form. HE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS 11
The associated sheaf J mf d ( X ), denoted by W ( X ) in [11], consists of maps( π, f, j ) : E −→ X × R × R d − ∞ with(a) ( π, f ) is proper map,(b) ( f, j ) is an embedding,(c) π : E −→ X is a submersion of relative dimension d ,(d) for x ∈ X , f x : E x → R is “Morse”, i.e. its 2-jet satisfies the conditions (i), (ii) and(iii).The space |J mf d | = |W| was determined up to homotopy in Theorems 1.2 and 3.5 of [11]. Werecall the results. Let G mf ( d, n ) be the space of pairs( V, f ) ∈ G ( d, n ) × J ( V )with f satisfying the above conditions (i), (ii) and (iii) and f (0) = 0. Let ˆ U ⊥ d,n be thecanonical n -dimensional bundle over G mf ( d, n ) and M T mf ( d ) be the spectrum with M T mf ( d ) d + n = Th ( ˆ U ⊥ d,n ) . Theorem 3.5. ([MW])
There is a homotopy equivalence Ω |J mf d | ∼ = Ω ∞ M T mf ( d ) := colim n Ω d + n Th ( ˆ U ⊥ d,n ) . Analogous to (11), there is the cofibration of spectra(17) Σ − M T ( d − −→ M T mf ( d ) −→ Σ ∞ (Σ mf ( d, ∞ ) + )The inclusion J mf d ( X ) −→ J gmf d ( X ) induces a mapΩ |J mf d | −→ Ω |J gmf d | of moduli spaces which can be examined upon comparing the (11) and (17) We have thehomotopy commutative diagram of homotopy fibrations(18) Ω ∞ Σ − M T ( d −
1) Ω ∞ M T mf ( d ) Ω ∞ Σ ∞ (Σ mf ( d, n ) + )Ω ∞ Σ − M T ( d −
1) Ω ∞ M T gmf ( d ) Ω ∞ Σ ∞ (Σ gmf ( d, n ) + ) ❄ ∼ = ✲ ❄ ✲ ❄✲ ✲ The middle vertical row can be identified with the mapΩ |J mf d | −→ Ω |J gmf d | and the right-hand vertical row corresponds to the right-hand arrow of Proposition 3.3. Thisgives Corollary 3.6.
There is a homotopy fibration d − Y i =0 Ω ∞ Σ − ( BO ( i ) × BO ( d − i − + −→ Ω |J mf d | −→ Ω |J gmf d | . ✷ Generalization to tangential structures.
Let θ : B → BO ( d ) be a Serre fibrationthought as a structure on d -dimensional vector bundles : If f : X → BO ( d ) is a map classifyinga vector bundle over X , then a map ℓ : X → B such that f = θ ◦ ℓ . For a given n , we definethe space G θ, gmf ( d, n ) as the pull-back:(19) G θ, gmf ( d, n ) BG gmf ( d, n ) BO ( d ) ✲❄ θ d,n ❄ θ ✲ i n where i n is the composition of the forgetful map G gmf ( d, n ) → G ( d, n ) and the canonical em-bedding i on : G ( d, n ) ֒ → G ( d, ∞ ) = BO ( d ). To define a corresponding sheaf J θd of generalizedMorse function on manifolds with tangential structure θ , we use Definition 2.1 but adding therequirement that the manifold E and corresponding fibers E x = π − ( x ) are equipped withthe compatible tangential structures. Similarly the sheaf h J θd is well-defined and there is thecorresponding map j π ( θ ) : |J θd | −→ | h J θd | . The following result is a generalition of Theorem2.2 providing the h -principle: Theorem 3.7.
The map j π ( θ ) : |J θd | −→ | h J θd | is weak homotopy equivalence. ✷ To describe the homotopy type of the moduli space Ω |J θd | , we consider the bundle diagram: V θ, ⊥ d,n V ⊥ d,n G θ, gmf ( d, n ) G gmf ( d, n ) ✲❄ ❄✲ θ d,n where V θ, ⊥ d,n is a pull-back of V ⊥ d,n . Similarly, let V θd,n → G θ, gmf ( d, n ) be the pull-back of thebundle V d,n → G gmf ( d, n ). The Thom space of the bundle V θ, ⊥ d,n gives rise to the spectrum M T θ, gmf ( d ) which in degrees ( d + n ) is(20) M T θ, gmf ( d ) d + n = Th ( V θ, ⊥ d,n ) . We have the following version of Theorem 3.1:
Theorem 3.8.
There is weak homotopy equivalence Ω | h J θd | ∼ = Ω ∞ M T θ, gmf ( d ) . ✷ HE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS 13
We next examine the homotopy type of Ω ∞ M T θ, gmf ( d ) in terms of the corresponding singularsets Σ θ, gmf ( d, n ) ⊂ G θ, gmf ( d, n ). Define the θ -Grassmannian G θ ( d, n ) as the pull-back:(21) G θ ( d, n ) BG ( d, n ) BO ( d ) ✲❄ θ n ❄ θ ✲ i on where i on : G ( d, n ) ֒ → BO ( n ) is the canonical embedding, G θ ( d, n ) = { ( V, b ) | i on ( V ) = θ ( b ) } ⊂ G ( d, n ) × BO ( d ) . Then it is easy to identify G θ, gmf ( d, n ) with the subspace G θ, gmf ( d, n ) = { ( V, b, f ) | ( V, b ) ∈ G θ ( d, n ) , f ∈ J ( V ) } . Let Σ θ, gmf ( d, n ) be the singular set of triples ( V, b, f ), where f : V → R has vanishing linearterm. The non-sigular subspace G θ, gmf ( d, n ) \ Σ θ, gmf ( d, n ) is the set of triples ( V, b, f ) with f = ℓ + q + r , where the linear part ℓ = 0. By analogy with the space ˆ G gmf ( d, n ) from Lemma3.2, we define ˆ G θ, gmf ( d, n ) = { ( V, b, f ) | f = ℓ + q + r, | ℓ | = 1 , q = 0 , r = 0 } and notice that the space G θ, gmf ( d, n ) \ Σ θ, gmf ( d, n ) retracts to ˆ G θ, gmf ( d, n ). By construction,the space ˆ G θ, gmf ( d, n ) is the pull-back in the diagram:ˆ G θ, gmf ( d, n ) ˆ G gmf ( d, n ) G θ ( d, n ) G ( d, n ) ✲❄ ❄ g n ✲ θ n where θ n : G θ ( d, n ) → G ( d, n ) is from (21) and g n is a composition of the forgetting map andthe inclusion: g n : ˆ G gmf ( d, n ) ֒ → G gmf ( d, n ) → G ( d, n )Similarly to the above case, the normal bundle of the inclusion Σ θ, gmf ( d, n ) ֒ → G θ, gmf ( d, n )coincides with ( V θd,n ) ∗ ∼ = V θd,n restricted to Σ θ, gmf ( d, n ), and again the inclusion( D ( V θd,n ) , S ( V θd,n )) −→ ( G θ, gmf ( d, n ) , G θ, gmf ( d, n ) \ Σ θ, gmf ( d, n ))is an excision map. Let j θ : G θ, gmf ( d, n ) \ Σ θ, gmf ( d, n ) → G θ, gmf ( d, n ) be the inclusion. Thisleads to the cofibration(22) Th ( j ∗ θ V θ, ⊥ d,n ) → Th ( V θ, ⊥ d,n ) → Th (( V θ, ⊥ d,n ⊕ V θd,n ) Σ θ, gmf ( d,n ) )and to the cofibration of spectraΣ − M T θ ( d − → M T θ, gmf ( d ) → Σ ∞ (Σ θ, gmf ( d, ∞ ) + ) with corresponding homotopy fibration of infinite loop spaces:Ω ∞ Σ − M T θ ( d − → Ω ∞ M T θ, gmf ( d ) → Ω ∞ Σ ∞ (Σ θ, gmf ( d, ∞ ) + ) . We denote by C ob θ the corresponding cobordism category of manifolds equipped with tan-gential structure θ and by C ob θ, gmf the category with the condition that each morphism beequipped with a generalized Morse function as above. Corollary 3.9.
There is weak homotopy equivalence B C ob θ, gmf d ≃ Ω ∞− M T θ, gmf ( d ) , and the forgetful map B C ob θ, gmf d → B C ob θd induces isomorphism Ω θ, gmf d = π B C ob θ, gmf d ∼ = π B C ob d = Ω θd , where Ω θ, gmf d and Ω θd are corresponding cobordism groups. References [1] R. Cohen, A decomposition of the space of generalized Morse functions. Algebraic topology and algebraic K -theory (Princeton, N.J., 1983), 365–391, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton,NJ, 1987.[2] Y. Eliashberg, N. Mishachev, Wrinkling of smooth mappings and its applications. I, Invent. Math. 130,1997, no. 2, 345–369.[3] Y. Eliashberg, N. Mishachev, Wrinkling of smooth mappings. II. Wrinkling of embeddings and K. Igusa’stheorem, Topology, 39, 2000, no. 4, 711–732.[4] S. Galatius, Ib Madsen, U. Tilliman, M. Weiss, The cobordism type of the cobordism category, ActaMath., 202 (2009), 195-239.[5] M. Gromov, Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9.Springer-Verlag, Berlin, 1986. x+363 pp.[6] K. Igusa, Higher singularities of smooth functions are unnecessary. Ann. of Math. (2) 119 (1984), no. 1,1–58.[7] K. Igusa, On the homotopy type of the space of generalized Morse functions. Topology 23 (1984), no. 2,245–256.[8] K. Igusa, The stability theorem for smooth pseudoisotopies, K-Theory 2 (1988), no. 1–2, vi+355.[9] K. Igusa, C local parametrized Morse theory, Topology Appl. 36 (1990), no. 3, 209–231.[10] K. Igusa, Higher Franz-Reidemeister Torsion, AMS/IP Studies in Advance Mathematics, vol. 31, Inter-national Press, 2002.[11] Ib Madsen, M. Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math.(2) 165 (2007), no. 3, 843–941.[12] A. Phillips, Submersions of open manifolds, Topology 6 (1967) 170–206[13] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications. Translationsof Mathematical Monographs, 98. American Mathematical Society, Providence, RI, 1992. vi+208 pp. HE MODULI SPACE OF GENERALIZED MORSE FUNCTIONS 15
Department of Mathematics, University of Oregon, Eugene, OR, 97403 USA
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