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Some Remarks on Cobordism of Maps on Locally OrientableWitt Spaces

JEAN-PAUL BRASSELET

ALICE K. M. LIBARDI

ELIRIS C.RIZZIOLLI

MARCELO J. SAIA

Aim of this work is to present some remarks on cobordism of normally non-singular maps between locally orientable Witt spaces. By using the Wu classesdefined by Goresky and Pardon in [12], we give also a definition of Stiefel-Whitneynumbers in this situation. Following Stong’s construction, we construct a map inthe respective intersection homology groups and we show in several cases thatnull-cobordism implies the vanishing of these Stiefel-Whitney numbers.

2010 Mathematics Subject Classification: Primary 57Q20; Secondary 55N33.

1 Introduction

R. Thom [20] was one of the first to define cobordism of smooth manifolds, hedetermined the structure of the oriented and the non-oriented cobordism algebras.Many authors worked on the subject, in particular Stong in his famous book [17] andin many papers. Let us quote also Atiyah [1] and Conner and Floyd [6].

Stong in [16] introduced and studied a notion of cobordism for maps f : X → Y ofclosed smooth manifolds. He defined Stiefel-Whitney numbers for a map and presenteda formula using cohomology groups with Z2 coefficients to prove that two maps arecobordant if and only if they have the same characteristic numbers. We remark thatall these results concern smooth manifolds and use classical homology theories. Theyare recalled in section 2. The natural question arises: What can we say in the caseof singular varieties? Among various works in this direction, we will quote those ofFriedman [7], Goresky and Pardon [12], Siegel [15], Sullivan [18] and Szucs [19].

On the one hand, in [12, Introduction] Goreky and Pardon discussed four classes ofsingular spaces for which they constructed characteristic classes numbers and such thatthese characteristic numbers determine the cobordism groups: orientable s-dualityspaces, orientable locally square-free spaces, locally orientable Witt spaces, locally

http://www.ams.org/mathscinet/search/mscdoc.html?code=2010 Mathematics Subject Classification: Primary 57Q20,(Secondary 55N33.)

2 Jean-Paul Brasselet, Alice K. M. Libardi, Eliris C.Rizziolli and Marcelo J. Saia

orientable spaces. They mention also other cobordism theories. Siegel in [15] describedthe class of Q-Witt spaces and computed the cobordism groups of such spaces, showingthat in non trivial cases they are equal to the Witt groups. Pardon [14] computed thecobordism groups of the “Poincare duality spaces" defined by Goresky and Siegel in[13]. Friedman in [7] follows Siegel by computing the bordism groups of oriented K -Witt spaces for any coefficient field K as well as identifying the resulting generalizedhomology theories.

On the other hand, in order to define cobordism for maps, Stong uses the constructionof a particular “Umkehrungs" homomorphism [16].

Our contribution, in this work, is to present some results in the vein of Stong. Weconsider pseudomanifolds which are locally orientable Witt spaces and normally non-singular maps. We observe that these conditions are strong ones, however they arenatural for the considered problem.

Our tracking is the following: On the one hand, characteristic classes of smoothmanifolds are defined in usual cohomology groups on which there is a multiplicativestructure given by the cup-product. On the other hand, characteristic classes of singularvarieties do not exist in cohomology, they lie in homology where there is no productstructure. In order to recover a product, one has to consider intersection homology.In general, characteristic classes cannot be lift to intersection homology (see [4]);fortunately the characteristic classes we consider lie in intersection homology. But,unlike homology theories, intersection homology is not homotopy invariant and doesnot, in general, satisfy an universal coefficient theorem. In this case, as we see in [7],the Witt spaces provide an important class of examples defined by a relatively tractablecondition concerning intersection homology.

Siegel in [15] described the class of Q-Witt spaces and computed the cobordismgroups of such spaces, showing that in non trivial cases they are equal to the Wittgroups. Pardon [14] computed the cobordism groups of the “Poincare duality spaces"defined by Goresky and Siegel in [13]. Friedman in [7] follows Siegel by computingthe bordism groups of oriented K -Witt spaces for any coefficient field K as well asidentifying the resulting generalized homology theories.

The choice of normally nonsingular maps is natural for our considerations: they havethe properties we need. In particular, if two such maps are cobordant in our sense,then we have coincidence of the Stiefel-Whitney numbers defined using the Wu classesgiven by Goresky and Pardon in [12]. In this context we show in several cases that, iftwo maps are cobordant, then they have the same characteristic numbers, consequentlythese numbers are cobordism invariants. To prove these results we show that if f is

Some Remarks on Cobordism of Maps on Locally Orientable Witt Spaces 3

null-cobordant then these numbers are zero. Firstly we consider the case X is a locallyorientable Witt space of pure dimension a and Y an b-dimensional smooth manifold.Then we consider the case X is an a -dimensional smooth manifold and Y a locallyorientable Witt space of pure dimension b. To conclude, we consider the general casewhere X and Y are locally orientable Witt spaces. As a consequence we obtain thatthe characteristic numbers defined in the described cases are cobordism invariants.

Acknowledgement: The first named author thanks CNPq, Bolsa Pesquisador VisitanteEspecial Process: 400580/2012-8 and FAPESP Process: 2015/06697-9. The secondnamed author thanks FAPESP Process: 2012/24454-8. The third and fourth namedauthors thank FAPESP Process: 2014/00304-2 and CNPq Processes: 482183/2013-6and 300733/2009-7.

2 Results of Stong and Atiyah on cobordism of maps used inthis work

Let us recall the Stong resuts in [16, §2] and the relationship with the Atiyah bordismgroups.

A map of dimension (m, n) is a triple (f ,M,N) where M and N are closed smoothmanifolds of dimension m and n, respectively and f : M → N is a continuous map.

Definition 2.1 Two maps (f ,M,N) and (f′,M′,N′) of dimension (m, n) are cobordant

if there exists a triple (F,V,W) where:

(1) V and W are compact smooth manifolds of dimensions m + 1 and n + 1respectively with boundaries ∂V = M ∪M

′and ∂W = N ∪ N

′, and

(2) F : V → W is a continuous map whose restrictions to M and N are f and f′

respectively.

The set of equivalence classes of maps of dimension (m, n) under this relation isdenoted N (m, n). Following Thom, Stong shows how to reduce the calculation of thecobordism groups N (m, n) to a homotopy question involving the Thom spaces. Therelation with the bordism groups Nm(N), introduced by Atiyah, is given by:

Proposition 2.2 Let (f ,M,N) and (f′,M′,N′) be maps of dimension (m, n) having the

same target N = N′. If two maps represent the same class in Nm(N), they determine

the same class in N (m, n).


Corollary 2.3 If f , g : M → N are homotopic maps, then (f ,M,N) is cobordant to(g,M,N).

Stong defines the Stieffel-Whitney (S-W for short) numbers associated to the map(f ,M,N) and proves that two maps of the same dimension (m, n) are cobordant ifand only if they have the same S-W numbers. That is, he proves that the resultingcobordism groups N (m, n) are determined by the S-W numbers. In order to prove thisresult, Stong uses a “Umkehrungs" homomorphism f ! that we define below.

Definition 2.4 [16, §6] Let us consider a map f : M → N , where M and N aremanifolds of dimensions m and n, respectively. One defines a map f ! : Hi(M) →Hi+n−m(N) in such a way that for any α ∈ Hi(M), the map f !(α) : Hi+n−m(N) → Z2

satisfies: for each β ∈ Hi+n−m(N),

f !(α)(β) = 〈f ∗(β) ∪ α, [M]〉 ∈ Z2,

where β ∈ Hm−i(N) is the Poincare dual of β .

Remark 2.5 According to Atiyah and Hirzebruch [2], the map f ! can be also describedin the following way: let us consider h : M → Ss an imbedding of M in some s-dimensional sphere Ss and T a tubular neighborhood of (f × h)(M) in N × Ss , then f !

is the composition of the maps:

Hi(M)ϕ→ Hi+s+n−m(T/∂T) c∗→ Hi+s+n−m(N × Ss) ' Hi+n−m(N),

where ϕ denotes the Thom isomorphism and c : N × Ss → T/∂T is the collapsingmap.

Stong also proved that if n ≥ 2m then Ψ : N (m, n) −→ Nm(BO) ⊕ Nn defined byΨ(f ,M,N) = ([τ.f ,M], [N]) is an isomorphism, where τ : N −→ BO is the classifyingmap of the tangent bundle of N and Nn is the bordism group of n-dimensionalmanifolds, defined by Thom.

One of the reasons for which we restrict our study to normally nonsingular maps isthat, in this case, one can construct maps with analogous properties to those of the mapf ! described above.


3 Intersection homology

3.1 Pseudomanifolds

Definition 3.1 [12, 2.1] An a-dimensional pseudomanifold (without boundary) isa purely a-dimensional piecewise linear (P.L. for short) polyhedron which admits atriangulation such that each (a− 1) simplex is a face of exactly two a-simplices.

A pseudomanifold admits a pieciewise linear stratification, which is a filtration byclosed subspaces ∅ ⊂ X0 ⊂ X1 ⊂ . . . ⊂ Xa−2 ⊂ Xa = X , with the singular part Σ(X)of X being (included in) the element Xa−2 of the filtration and such that for each pointx ∈ Xi−Xi−1 there is a neighborhood U and a P.L. stratum preserving homeomorphismbetween U and Ra−i × c(L), where L is the link of the stratum Xi − Xi−1 and c(L)denotes the (open) cone on L . If Xi − Xi−1 is non empty, it is a (non necessarilyconnected) manifold of dimension i, and is called the i-dimensional stratum of thestratification.

Definition 3.2 [12, 2.3] A pseudomanifold X with boundary ∂X is an a-dimensionalcompact P-L space such that X− ∂X is a pseudomanifold and ∂X is a compact a− 1-dimensional P.L. subspace of X which has a collared neighborhood U , i.e. there is aP.L. homeomorphism ϕ : U ∼= ∂U× [0, 1) such that the restriction ϕ|∂X → ∂X×0is the identitiy.

Definition 3.3 [8] A map f : X → Y between pseudomanifolds is normally nonsin-gular if there exists a diagram

Nf

π

i // Y × Rk

p

X

s

TT

f // Y

where π : Nf −→ X is a vector bundle with zero-section s, i is an open embedding,p is the first projection and one has f = p i s. The bundle Nf is called the normalbundle.

3.2 Intersection Homology and Cohomology

All homology and cohomology groups will be considered with Z2 coefficients. Refer-ence for this section is Goresky-MacPherson original paper [10].


The notion of perversity is fundamental for the definition of intersection homology andcohomology. A perversity p is a multi-index sequence of integers (p(2), p(3), . . .) suchthat p(2) = 0 and p(c) ≤ p(c+1) ≤ p(c)+1, for c ≥ 2. Any perversity p lies betweenthe zero perversity 0 = (0, 0, 0, . . .) and the total perversity t = (0, 1, 2, 3, . . .). Inparticular, we will use the lower middle perversity, denoted m and the upper middleperversity, denoted n, such that

m(c) =

[c− 2

2

]and n(c) =

[c− 1

2

], for c ≥ 2.

Let p be a perversity, the complementary perversity q is defined by

q(c) + p(c) = t(c) = c− 2 for all c ≥ 2.

Let p and r be two perversities, if, for every c ≥ 2, one has p(c) ≤ r(c), one will writep ≤ r .

Let X be an a-dimensional pseudomanifold and p a perversity. The intersectionhomology groups with Z2 coefficients, denoted IHp

i (X), are the homology groups ofthe chain complex

ICpi (X) =

ξ ∈ Ci(X) | dim(|ξ| ∩ Xa−c) ≤ i− c + p(c) and

dim(|∂ξ| ∩ Xa−c) ≤ i− 1− c + p(c), ∀c ≥ 2.

,

where Ci(X) denotes the group of compact i-dimensional P.L. chains ξ of X with Z2

coefficients and |ξ| denotes the support of ξ .

The intersection cohomology groups with Z2 coefficients, denoted IHa−ip (X), are

defined as the groups of the cochain complex

ICa−ip (X) =

γ ∈ Ca−i(X) | dim(|γ| ∩ Xa−c) ≤ i− c + p(c) and

dim(|∂γ| ∩ Xa−c) ≤ i− 1− c + p(c), ∀c ≥ 2.

,

where Ca−i(X) denotes the abelian group, with Z2 coefficients, of all (a−i)-dimensionalP.L. cochains of X with closed supports in X .

The main properties of intersection homology that we will use are the following:

For any perversity p, the Poincare map PD, cap-product by the fundamental class ofX , naturally factorizes in the following way [10]:

Ha−i(X) PD //

αX

%%

Hi(X)

IHpi (X),

ωX

::


where αX is induced by the cap-product by the fundamental class [X] and ωX isinduced by the inclusion ICp

i (X) → Ci(X).

For perversities p and r such that p + r ≤ t , the intersection product

IHpi (X)× IH r

j (X; )→ IHp+r(i+j)−a(X)

is well defined.

The natural homomorphism IHa−ip (X)→ IHp

i (X), cap product by the fundamental class[X], is an isomorphism.

4 Locally orientable Witt spaces and Wu classes

In this section we use definitions and notations of M. Goresky [9] and M. Goresky andW. Pardon [12]. First of all, let us fix notations in the smooth case.

Let M be an m dimensional manifold. We will denote by wi(M) ∈ Hi(M) the Stiefel-Whitney cohomology classes (S-W cohomology classes) of the tangent bundle TM .The Stiefel-Whitney homology classes (S-W homology classes) of TM denoted bywm−i(M) in Hm−i(M) are their images by Poincare duality.

For j : M → V the inclusion map of smooth manifolds, one has the naturality formulaj∗(wi(V)) = wi(M).

4.1 Z2 -Witt spaces and Wu classes

In the singular case, the Steenrod square operations are defined in intersection coho-mology by M. Goresky [9, §3.4] as follows:

Definition 4.1 Let X be an a-dimensional pseudomanifold. Suppose c and d areperversities such that 2c ≤ d . For any i with 0 ≤ i ≤ [a/2] the “Steenrod square”operation

Sqi : IH`c(X)→ IHi+`

d (X)→ Z2

is given by multiplication with the intersection cohomology ith -Wu class of X :

vi(X) = vid(X) ∈ IHi

d(X).

One defines vi(X) = 0, for i > [a/2].


Definition 4.2 ([12, Definition 10.1]) A stratified pseudomanifold X is a Z2 -Wittspace if, for each stratum of odd codimension 2k + 1, then IHn

k (L) = 0, where L is thelink of the stratum.

Let X be a Z2 -Witt space, then the Wu classes vi(X) lift canonically to IHim(X) =

IHin(X) (see [12] §10). We denote by va−i(X) ∈ IHn

a−i(X) the (homology) (a − i)th -Wu class of X , in intersection homology, dual to the Wu class vi(X) (denoted byIvi ∈ IHi(X) in [9]).

For such spaces, the middle intersection homology group satisfies the Poincare dualityover Z2 .

Definition 4.3 [9, 15] One defines the Whitney classes by

Iwa−i(X) =∑`+j=i

Sq`vj(X) ∈ IHit(X) = Ha−i(X).

The pullback of the intersection cohomology Whitney class under a normally nonsin-gular map is given by the following theorem ([9, §5]):

Theorem 4.4 Let X and Y be Z2 -Witt spaces and f : X → Y a normally nonsingularmap with normal bundle Nf . Then one has, in IH∗(X):

f ∗(Iw(Y)) = ω(Nf ) ∪ Iw(X)

where ω(Nf ) is the Whitney cohomology class (in H∗(X)) of the normal bundle Nf .

The inclusion j : X → V provides an unique morphism j∗ : IHb−iq (V) → IHb−i

q (X)(see [4] §(3.4)). The result comes from the commutative diagram

IHb−iq (X) IHb−i

q (V)j∗oo

IHpi (X)

∼=

OO

IHpi+1(V)

j∗Xoo

∼=

OO

where the bottom map j∗X is defined by the upper one and q is the complementaryperversity of p. When X = ∂V , the inclusion is a normally nonsingular map, so onehas the corollary.

Corollary 4.5 If j : X → V is the inclusion map from X into V , where X = ∂V ,then one has:

j∗X(vi+1(V)) = vi(X).


4.2 Locally orientable Witt spaces

Definition 4.6 ([12, 8.1]) A stratified pseudomanifold X is locally orientable if, foreach stratum, the link is an orientable pseudomanifold.

Definition 4.7 ([12, 10.2]) A stratified pseudomanifold X is a locally orientable Wittspace if it is both locally orientable and a Z2 -Witt space.

Lemma 4.8 ([12, 10.2]) If X is a locally orientable Witt space then Sq1Sq2i = Sq2i+1

as homomorphismsIHm

j (X,Z2)→ IHmj−2i−1(X,Z2).

In our proofs, we will use the following property of locally orientable Witt spaces(see [12]): In the situation of a locally orientable Witt space, the Wu classes whichare defined as middle intersection homology classes, can be muliplied to construct thecharacteristic numbers

vi(X)vj(X) = (v1(X))n−i−j.

Cohomology Wu classes can be multiplied with homology Wu classes to producecharacteristic numbers.

We notice that “absolute" cobordism of locally orientable Witt spaces is computed in[12, 10.5].

5 Cobordism of normally nonsingular maps

Definition 5.1 Let f : X → Y be a normally nonsingular map between pseudomani-folds of dimensions a and b respectively. The triple (f ,X,Y) is null-cobordant if thereexist:

(1) Pseudomanifolds V and W with dimensions a + 1 and b + 1, respectively, and∂V = X and ∂W = Y .

(2) F : V → W normally nonsingular map such that the following diagram com-mutes

UX

∼= φ

F|UX // UY

∼=ψ

∂V × [0, 1)f×Id // ∂W × [0, 1),


where Ux and UY are the collared neighborhood of X and Y in V and W respec-tively, where φ and ψ are P.L. diffeomorphisms such that φ(x) = (x, 0), x ∈ ∂Vand ψ(y) = (y, 0), y ∈ ∂W .

(3) F|∂V = f : ∂V → ∂W.

Given normally nonsingular maps f : X1 → Y1 and g : X2 → Y2 , one may define(f t g,X1 t X2,Y1 t Y2) by mapping the disjoint union of X1 and X2 into the disjointunion of Y1 and Y2 by the map such that (f t g)|X1 = f and (f t g)|X2 = g.

Definition 5.2 Two normally nonsingular maps f : X1 → Y1 and g : X2 → Y2 arecobordant if the triple (f t g,X1 t X2,Y1 t Y2) is null-cobordant.

Lemma 5.3 Let us fix the image set Y , the cobordism relation defined in 5.2 in the setof normally nonsingular maps f : X → Y between pseudomanifolds is an equivalencerelation.

Proof In order to prove that (f ,X,Y) is cobordant to itself, let us consider V = X× I ,W = Y × I with standard collar neighborhoods and define F : V → W by F(x, t) =

(f (x), t), t ∈ I, x ∈ X . Remarking that ∂(X × I) = ∂X × I ∪ X × ∂I = X × 0 ∪ X × 1and one has the same for ∂(Y × I), implies that F|∂V = f ∪ f and reflexivity. Letus prove the symmetry. If (f1,X1,Y) is cobordant to (f2,X2,Y) by (F,V,W), then Vand W with opposite collar structure provides the result. In order to prove transitivity,assume (f1,X1,Y) is cobordant to (f2,X2,Y), then there exist V1 with ∂V1 = X1∪−X2 ,where −X2 denotes X2 with opposite orientation and define W1 = Y × (−1, 1). Alsothere exist UX1 ,UX2 collar neighborhoods of X1 in V1 and of X2 in V2 respectively,and collar neighborhood W1 of Y and a map F1 : V1 → W1 satisfying the definitionof cobordism. Analogous from (f2,X2,Y) cobordant to (f3,X3,Y) one has V2 with∂V2 = X2 ∪ −X3 and W2 = Y × (−1, 1) with collar neighborhoods U

′X2,UX2 of X2

in V2 and X3 in V3 respectively, and collar neighborhoods of Y in W2 . One takesV = V1 ∪ V2 , with V1 ∩ V2 = X2 , W = W1 ∪W2 where the gluing is done using thecollars and F = F1 ∪ F2 then obtaining the transitivity.

When fixing the target space Y , we denote the group of cobordism of maps f : X → Ybetween pseudomanifolds by Sηm(Y), the operation being given by the disjoint union.

The definition of relative cobordism groups can be analogously to the one of absolutecobordism group.


Definition 5.4 Let us fix (Y,B). A map f : (X, ∂X) → (Y,B) is null-cobordant ifthere are V and W with X ⊂ ∂V and Y ⊂ ∂W , and a normally nonsingular mapF : V → W such that F|∂V = f : ∂V → ∂W and F(∂V − X) ⊂ B and the followingdiagram commutes:

UX

∼= φ

F|UX // UY

∼=ψ

∂V × [0, 1)f×Id // ∂W × [0, 1),

where Ux and UY are the collared neighborhood of X and Y respectively, and φ and ψare P.L. diffeomorphisms such that φ(x) = (x, 0), x ∈ ∂V and ψ(y) = (y, 0), y ∈ ∂W .

We note that relative cobordism of maps f : (X,A) → (Y,B) is also a relation ofequivalence, the quotient group being denoted by Sηm(Y,B) and the operation beingthe disjoint union.

5.1 Eilenberg-Steenrod Axioms

Let h : (X,A) → (Y,B) be a continuous map. We consider the induced map h∗ :Sη∗(X,A) → Sη∗(Y,B) defined by h∗(f ,M,X) = (hf ,M,Y) and the boundary map∂ : Sηn(X,A)→ Sηn−1(A), defined by ∂(f ,M,X) = (f |∂M, ∂M,A).

The following Eilenberg-Steenrod axioms are satisfied:

1. If id : (X,A)→ (X,A) is the identity map, then id∗ : Sη∗(X,A)→ Sη∗(X,A) is theidentity isomorphism,

2. If φ : (X1,A1)→ (X2,A2) and θ : (X2,A2)→ (X3,A3) are maps, then (φθ)∗ = θ∗φ∗ ,

3. For any map φ : (X,A)→ (Y,B), one has: ∂(φ |)∗A = (φ)∗∂ ,

4. For φ, ψ : (X,A)→ (Y,B) homotopic maps, one has φ∗ = ψ∗ ,

5. There exists a long exact sequence

. . .δ→ Sηn(A) i∗→ Sηn(X)

j∗→ Sηn(X,A) δ→ Sηn−1(A)→ . . .

where i∗ and j∗ are induced by inclusions of pairs

i : (A, ∅)→ (X, ∅) and j : (X, ∅)→ (X,A)

andδ : Sηn(X,A)→ Sηn−1(A)


is the boundary map,

6. If U ⊂ int(A) , then the inclusion map i : (X − U,A − U) → (X,A) induces anisomorphism i∗ : Sη∗(X − U,A− U)→ Sη∗(X,A).

Then these cobordism groups fit together into a generalized homology theory.

Remark 5.5 In the smooth case, there is an isomorphism between N∗(Y) andH∗(Y,Z2) ⊗ N∗ where N∗(Y) is the group of non-orientable bordism and N∗ isthe Thom bordism group [6].

A natural question is to ask if there exists an isomorphism between Sη∗(Y) andIH∗(Y,Z2) ⊗ η∗Z2−Witt where η∗

Z2−Witt denotes the Z2 -Witt bordism group for thenon-oriented case. Friedman and Siegel defined the Z2 -Witt bordism group Ω∗

Z2−Witt

for the oriented case; they do not consider the case of maps between Z2 -Witt spaces.

We are not able to prove an isomorphism between Sη∗(Y) and IH∗(Y,Z2)⊗ η∗Z2−Witt ,however one of the ingredients to explicit such isomorphism would be to show that anyclass α ∈ IHm

i (X;Z2) can be represented by a (singular) variety B ⊂ X . We prove thisresult considering the intersection homology group with coefficients in Q.

Theorem 5.6 Let α ∈ IHmi (X;Q), then there exists an i-dimensional variety B such

that f!([B]) = α where f! : IHmi (B;Q) → IHm

i (X;Q) is a lifting of the inclusionf : B → X .

Proof Let [α] ∈ IHmi (X;Q) and consider the image [ωX(α)] ∈ Hi(X;Q), by the map

ωX : IHmi (X;Q) → Hi(X;Q). For a suitable triangulation of X , the class [ωX(α)] can

be represented by a (simplicial) cycle B, whose support will also be denoted by B,and its fundamental class denoted by [B] ∈ Hi(B;Q). Considering the stratificationof B induced by the one of X , one has a map ωB : IHm

∗ (B;Q) → H∗(B;Q). Thefundamental class [B] ∈ Hi(B;Q) is represented by the allowable cycle B, so it can belifted in a class also denoted by [B] ∈ IHm

i (B;Q).

On the other hand, the graph of the inclusion f : B → X is the element (B, f (B)) =

(B, ωX(α)) ∈ Hi(B× X;Q). The graph admits a lifting (B, α) ∈ IHmi (B× X;Q) by the

map ω(B×X) : IHmi (B× X;Q)→ Hi(B× X;Q).

By [4], there is a one-to-one correspondence between lifting of the graph (B, f (B)) inIHm

i (B×X;Q) and maps f! : IHmi (B;Q)→ IHm

i (X;Q) such that the following diagram


commutes:IHm

i (B;Q)

f!

ωB // Hi(B;Q)

f∗

IHmi (X;Q)

ωX // Hi(X;Q).

More precisely, the lifting (B, α) ∈ IHmi (B×X;Q) corresponds to a map f! : IHm

i (B;Q)→IHm

i (X;Q) such that f!([B]) = α .

6 Main results

In this section we consider different cases of pseudomanifolds X and Y and normallynonsingular maps f : X → Y and prove that if (f ,X,Y) is null-cobordant, then the S-Wnumbers associated in each case are zero. Firstly we consider the case X is a locallyorientable Witt space of pure dimension a and Y a b-dimensional smooth manifold.Then we consider the case X is an a-dimensional smooth manifold and Y is a locallyorientable Witt space of pure dimension b. To conclude, we consider the general casewhere X and Y are locally orientable Witt spaces.

6.1 Case of a map f : X −→ Y , with Y a smooth manifold

Let f : X −→ Y be a normally nonsingular map, with X a locally orientable Witt spaceof pure dimension a and Y a b-dimensional smooth manifold.

Definition 6.1 Let us define the map fB : IH p

i (X) → IH p

i (Y) in such a way that thefollowing diagram commutes

Hi(X)f∗ // Hi(Y)

IHpi (X)

ωX

OO

fB // IHpi (Y)

ωY '

OO

i.e. fB = (ωY )−1 f∗ ωX, where the map ωY is an isomorphism since Y is smooth.

We denote by fB the map obtained by composition

IHpi (X) ωX−→ Hi(X)

f∗−→ Hi(Y) PD−→ Hb−i(Y)

with Poincare duality PD.


Definition 6.2 For any partition ` = (`1, . . . , `s) and r numbers u1, . . . , ur satisfying

(1) (`1 + `2 + . . . `s) + u1 + . . .+ ur + r(b− a) = b,

let us denote ω`(Y) = ω`1(Y) · · ·ω`s(Y). The S-W numbers of any triple (f ,X,Y) aredefined by

〈ω`(Y).fB(va−u1(X)). · · · .fB(va−ur (X)), [Y]〉.

Theorem 6.3 Let f : X −→ Y be an normally nonsingular map, with X a locallyorientable Witt space of pure dimension a and Y a b-dimensional smooth manifold. If(f ,X,Y) is null-cobordant, then for any partition ` and r numbers u1, . . . , ur satisfying(1), the S-W numbers

〈ω`(Y).fB(va−u1(X)). · · · .fB(va−ur (X)), [Y]〉

are zero.

Proof As (f ,X,Y) is cobordant, one has (f ,X,Y) = ∂(F,V,W). We may define amap

FB : IHpi (V)→ IHp

i (W) = Hi(W)→ Hb+1−i(W)

in the same way that we defined fB .

One has:〈ω`(Y).fB(va−u1(X)). · · · .fB(va−ur (X)), [Y]〉 =

〈j∗w`(W).j∗FB(va−u1(V)). · · · .j∗FB(va−ur (V)), ∂[W]〉,

by corollary 4.5 and commutativity of the following diagram:

Hi+1(V)F∗ //

j

%%

Hi+1(W)j

xxHi(X)

f∗ // Hi(Y)

IHpi (X)

ωX

OO

fB // Hb−i(Y)

PD ∼=

OO

IHi+1(V)FB //

j∗X99

ωV

OO

Hb−i(W).

j∗ee

PD ∼=

OO

So, we obtain:⟨j∗(ω`(W).FB(va−u1(V)). · · · .FB(va−ur (V))

), ∂[W]

⟩=


⟨δj∗(ω`(W).FB(va−u1(V)). · · · .FB(va−ur (V))

), [W, ∂W]

⟩= 0,

whereHk(W)

j∗→ Hk(Y) δ→ Hk+1(W, ∂W)

is part of a long exact sequence, so that δj∗ = 0.

6.2 Case of a map f : X −→ Y , with X a smooth manifold

Let f : X −→ Y be a normally nonsingular map, with X an a-dimensional smoothmanifold and Y a locally orientable Witt space of pure dimension b.

Since f is a normally nonsingular map one may consider the normal bundle Nf overX , and i : Nf → Y × Rs+1 an open imbedding. Let T be a tubular neighborhoodof (f × h)(X) in Y × Rs+1 , where h : X → Rs+1 is defined in such a way that thefollowing diagram commutes.

Nfi // Y × Rs+1

X

σ

OO

f×h

::

We denote by Ss the s-dimensional sphere in Rs+1 and by T the intersection T =

T ∩ (Y × Ss).

Following the remark 2.5, there exists a map φ which is the composition of the maps:

Hi(X)ϕ→ Hi+s+b−a(T/∂T) c∗→ Hi+s+b−a(Y × Ss) ' Hi+b−a(Y),

here ϕ denotes the Thom homomorphism and c : Y × Ss → T/∂T is the collapsingmap. The last isomorphism is given by the Kunneth formula for a product of a smoothmanifold with a Z2 - Witt space [5].

Since X is a smooth manifold, αX : Hi(X) → IHpa−i(X) is an isomorphism, then

one defines the map fB by commutativity of the following diagram, i.e. as beingfB = αY φ α−1

X

Hi(X)φ //

αX ∼=

Hb−(a−i)(Y)

αY

IHpa−i(X)

fB // IHpa−i(Y).


For any u with 0 ≤ u ≤ b, let vu(Y) ∈ IHmu (Y) the Wu class of Y , dual of vb−u(Y) ∈

IHb−um (Y) and wb−u(X) the homology Whitney class of X , so that fB(wb−u(X)) ∈

IHmb−u(Y). For any u with 0 ≤ u ≤ b the S-W intersection numbers

vu(Y) . fB(wb−u(X))

are well defined.

Theorem 6.4 Let f : X −→ Y be a normally nonsingular map, with X an a-dimensional smooth manifold and Y a locally orientable Witt space of pure dimensionb. If (f ,X,Y) is null-cobordant, then for any 0 ≤ u ≤ b the S-W numbers

vu(Y) . fB(ωb−u(X))

are zero.

Proof If (f ,X,Y) = ∂(F,V,W), one has

Hi(V)ϕ→ Hi+s+b−a(T ′/∂T ′) c∗→ Hi+s+b−a(W × Ss) ' Hi+b−a(W),

where V is embedded in Ss and T ′ is a tubular neighborhood of (F × h)(V), whichgives rise to the corresponding map FB . Therefore we can consider the followingdiagram, where PD denotes the Poincare duality

Ha−i(X)f∗ // Ha−i(Y)

IHpa−i(X)

ωX

OO

fB // IHpa−i(Y)

ωY

OO

Hi(X)

PD'

@@

ϕ //

αX

88

// Hi+s+b−a(T/∂T) c∗ // Hi+s+b−a(Y × Ss) ' Hi+b−a(Y)

αY

iiPD

dd

since we had defined fB and FB the result follows in the same way of the proof ofTheorem 6.3.

6.3 The general case

In the general case X and Y are locally orientable Witt spaces of dimensions a and brespectively. It is not always possible to define an unique map fB as done in the other


cases, however we can show that for any map fB considered, the bordism condition of(f ,X,Y) implies that the corresponding S-W numbers are zero.

First we show the following lemma.

Lemma 6.5 Let f : X −→ Y be a normally nonsingular map and (f ,X,Y) =

∂(F,V,W). Given a map fB there exists a map FB such that the following diagramcommutes.

IHmu (X)

fB

IHmu+1(V)

j∗Xoo

FB

IHm

u (Y) IHmu+1(W).

j∗Yoo

Proof The diagram

X

f

jX // V

F

Y jY // W

is a cartesian diagram. Then we can apply Proposition 10.7 in [3] (see also [11]).

One has equality of sheaves on Y :

j∗YF!A = f!j∗XA

for any sheaf A on V . That provides a commutative diagram of complexes of sheaveson Y (perverse intersection sheaves for the middle perversity m).

f!IC•

X

oo f!j∗X(IC•

V)

= j∗YF!(IC•

V)

IC•

Yoo j∗Y

(IC•

W)

Let us remind that intersection homology is obtained by taking hypercohomology ofthe perverse intersection sheaf:

IHmu (Y) = IHb−u(Y; IC•

Y )

Taking hypercohomologyIHb−u(Y; •)


of the previous diagram, one obtains:

IHb−u(X; IC•

X)

fB

ooj∗X IHb−u(V; IC•

V )

FB

IHb−u(Y; IC•

Y ) ooj∗Y IHb−u(W; IC•

W)

and the Lemma follows.

Theorem 6.6 Let f : X −→ Y be a normally nonsingular map, with X and Y locallyorientable Witt spaces of pure dimension a and b respectively. If (f ,X,Y) is null-cobordant, then for any u with 0 ≤ u ≤ b, the S-W numbers 〈vu(Y).fB(vb−u(X)), [Y]〉are zero.

Proof The diagram of Lemma 6.5 can be written in the cohomology setting

IHa−un (X)

f B// IHb−u

n (Y)

IHa−un (V)

j∗X

OO

FB// IHb−u

n (W).

j∗Y

OO

where m + n = t and we use the same notation for corresponding maps j∗X and j∗Y .

Let us consider the homology class vb−u(Y) ∈ IHnb−u(Y), that will be written vu(Y) in

IHum(Y) in the cohomology setting.

Then vu(Y) = j∗Yvu(W) where vu(W) ∈ IHum(W) is the corresponding cohomology Wu

class to the homology Wu class vb+1−u(W) ∈ IHnb+1−u(W) of W .

Let us consider the cohomology Wu class va−u(X) ∈ IHa−un (X) corresponding for

the cohomology Wu class vu(X) ∈ IHmu (X). Then va−u(X) = j∗X(va−u(V)) where

va−u(V) ∈ IHa−un (V) is the corresponding cohomology Wu class to the homology class

vu+1(V) ∈ IHmu+1(V).

One hasf B(va−u(X)) = f Bj∗X(va−u(X)) ∈ IHb−u

n (Y).

The intersection product

vb−u(Y) · fB(vu(X)) ∈ IHnb−u(Y)× IHm

u (Y)→ IH t0(Y)


corresponds to the product

vu(Y) ∪ f B(va−u(X)) ∈ IHum(Y)× IHb−u

n (Y)→ IHb0(Y).

One has

〈vu(Y) ∪ f B(va−u(X)), [Y]〉 =

〈j∗Yvu(W) ∪ f Bj∗X(va−u(V), [Y]〉 =

〈j∗Yvu(W) ∪ j∗YFB(va−u(V)), [Y]〉 =

〈j∗Y[vu(W) ∪ FB(va−u(V))

], [∂W]〉 =

〈δj∗Y[vu(W) ∪ FB(va−u(V))

], [W, ∂W]〉 = 0

where the first equality is a consequence of the Theorem 5.3 of Goresky [9], the secondone is from Lemma 6.5 and the fourth equality is obtained in an analogous way thanthe proof of Theorem 6.4.

From these results, it is possible to conclude that in each case above the S-W numbersdefined are cobordism invariants in the following sense.

Corollary 6.7 Let f : X1 → Y and g : X2 → Y be two cobordant normally nonsin-gular maps, then the corresponding characteristic numbers defined in each case aboveare equal.

References

[1] M.F.Atiyah.: Bordism and Cobordism, Proc. Ca.mb. Phil. Soc. (Math. Phys. Sci), 57,(1961), 200-208.

[2] M. F. Atiyah and F. Hirzebruch.: Cohomologie-Operationen und charaktieristischeKlassen, Maht. Z. 77 (1961), 149–187.

[3] A. Borel et al.: Intersection Cohomology, Progress in Mathematics, 50, Swiss seminars,Boston, (1984), ISBN: 0-8176-3274-3.


[4] G. Barthel, J.-P. Brasselet, K. Fieseler, O. Gabber and L. Kaup.: Relevement decycles algebriques et homomorphismes associes en homologie d’intersection [Liftingof algebraic cycles and associated homomorphisms in intersection homology], Ann. ofMath. 141, no 2, (1995), 147–179.

[5] D. C. Cohen, M. Goreski and L. Ji.: On the Kunneth for Intersection Cohomology.Trans. of the Amer. Math. Soc. 333 (1992), no. 1, 63 –69.

[6] P. E. Conner and E. E. Floyd.: Differentiable Periodic Maps, Springer, Berlin, (1964).

[7] G. Friedman .: Intersection homology with field coefficients: K-Witt spaces and K-Witt bordism, Communications on Pure and Applied Mathematics 62, (2009), no 9,1265-1292.

[8] W. Fulton and R. MacPherson.: Categorical framework for the study of singular spaces.Mem. Amer. Math. Soc., 31, (1981), no. 243.

[9] M. Goresky.: Intersection Homology operations, Comment. Math. Helvet. 59, (1984),485–505.

[10] M. Goresky and R. MacPherson.: Intersection homology theory, Topology 19 (1980),no. 2, 135–162.

[11] M. Goresky and R. MacPherson.: Intersection homology II, Invent. Math. 72 (1983),no. 1, 77–129.

[12] M. Goresky and W. Pardon.: Wu Numbers of Singular Spaces, Topology 28, (1989),325–367.

[13] M. Goresky and P. Siegel.: Linking pairings on Singular spaces, Comment. Math.Helvet. 58, (1983), 96–110.

[14] W. Pardon.: Intersection homology Poincare spaces and the characteristic varietytheorem, Comment. Math. Helvet. 65, (1990), 198–233.

[15] P. Siegel.: Witt Spaces: a Geometric Cycle Theory for KO-Homology at odd primes,Amer. J. Math., 105, (1983), no. 5, 1067–1105.

[16] R. E. Stong.: Cobordism of maps, Topology 5, (1966), 245–258.

[17] R. E. Stong.: Notes on Cobordism theory, Mathematical Notes, Princeton Univ. Press,(1968).

[18] D. Sullivan : Combinatorial invariants of analytic spaces. Proceedings of LiverpoolSingularities ï‰ Symposium I Springer Lecture Notes in Mathematics, vol. 192 of theseries pp 165-177.

[19] A. Szucs.: Cobordism of singular maps, Geom. Top. 12, (2008), 2379–2452.

[20] R. Thom.: Quelques proprietes globales des varietes differentiables, Comment. Math.Helvet. 28, (1954), 17–86.


I2M Aix-Marseille University, Marseille, France.

IGCE-UNESP, Rio Claro, S.P. Brasil.

IGCE-UNESP, Rio Claro, S.P. Brasil.

ICMC-USP, Sao Carlos, S.P. Brasil.

[email protected], [email protected],

[email protected], [email protected]

mailto:[email protected]




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