# A Bordism Viewpoint of Fiberwise Intersections

Authors: Gun Sunyeekhan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

We use the geometric data to define a bordism invariant for the fiberwise intersection theory. Under some certain conditions, this invariant is an obstruction for the theory. Moreover, we prove the converse of fiberwise Lefschetz fixed point theorem.


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## Body

## 1. Introduction

In topological fixed point theory, there is a classical questions that people may ask; given a smooth self-mapf:M→M of a smooth compact manifold M, when is f homotopic to a fixed point free map?The famous theorem, Lefschetz fixed point theorem, gave a sufficient condition to answer the above question as follows.Theorem 1 (Lefschetz fixed point theorem).
Letf:M→M be a smooth self-map of a compact smooth manifold M.
Iff has no fixed point, then the Lefschetz number L(f)=0, where L(f)=∑k≥0‍(-1)kTr(f*|Hk(M;ℚ)).In general, the converse of the above theorem does not hold. It requires a more refined invariant than the Lefschetz number to make the converse hold (see [1–3]).For this work, we focus on the similar arguments as above for the family of smooth maps over a compact base spaceB. The proof of the main theorem depends heavily on the intersection problem as follows.From now on, the notationsXa means the smooth manifold X of dimension a and I means the unit interval [0,1]. If Yb is a submanifold of Xa, f:X→Z, and η is a bundle over Z, then νY⊆X means the normal bundle of Y in X and f*(η) is a pull-back bundle of η along the map f.Later we define “framed bordism with coefficient in a bundle" as follows. LetX be a smooth manifold with a bundle ξ over it. Define Ω*fr(X;ξ) to be the bordism groups of manifolds mapping to X, together with a stable isomorphism of the normal bundle with the pullback of ξ. This framed bordism group will be a home for our invariants Lbord(f) (described in the last section) which detects more fixed point information than the regular Lefschetz number.(I) Suppose thatEPp+k, EQq+k, and EMm+k are smooth fiber bundles over a compact manifold Bk. Let f:EPp+k→EMm+k be a bundle map, and let EQ be a subbundle of EM with the inclusion bundle map iQ:EQ↪EM.We have a commutative diagram(1)
where P, Q, and M are the fibers of prP, prQ, and prM, respectively.We may assume thatf⋔EQ in EM (see [4]).The homotopy pullback is(2)E(f,iQ):={(x,λ,y)∈EP×EMI×EQ∣λ(0)=f(x),λ(1)=y}.We have a diagram which commutes up to homotopy(3)
where πP and πQ are the trivial projections; that is, we have a homotopy E(f,iQ)×I→KEM defined by K(x,λ,y,t)=λ(t), t∈I.We also have a mapc:f-1(EQ)→E(f,iQ) defined by x↦(x,cf(x),f(x)) where cf(x)= constant path in EM at f(x).Transversality yields a bundle map(4)Choose an embeddingEPp+k⊆SL, for sufficiently large L, where SL is a sphere of dimension L. Then, we have f-1(EQ)↪iEPp+k⊆SL. So νf-1(EQ)⊆SL≅νf-1(EQ)⊆EP⊕i*νEP⊆SL.The commutative diagram(5)
yields a bundle map
(6)Thus,(c,c^) determines an element [c,c^]∈Ωp+q+k-mfr(E(f,iQ);ξ).(II) Suppose that(N→c1E(f,iQ),νN⊆SL→c^1ξ) is another representative of [c,c^], where Np+q+k-m⊆EPp+k⊆SL. This means we have a normal bordism (W→𝒞E(f,iQ), νW→𝒞^ξ) between (c,c^) and (c1,c^1); that is (i)
Wp+q+k-m+1⊆(SL×I),(ii)
∂W⊆(SL×∂I),(iii)
W⋔(SL×∂I),(iv)
W∩(SL×0)=f-1(EQ) and W∩(SL×1)=Nsuch that
(7)𝒞|f-1(EQ)=c:f-1(EQ)⟶E(f,iQ),𝒞|N=c1:N⟶E(f,iQ),𝒞^|νf-1(EQ)⊆SL=c^,𝒞^|νN⊆SL=c^1.Theorem 2 (main theorem).
Assumem>q+((p+k)/2)+1. Then, there exists a smooth fiber-preserving map over B(8)Ψ:EP×I⟶EM
such that Ψ|EP×{0}=f, Ψ⋔EQ and Ψ|EP×{1}-1(EQ)=N.Note that if we letg=Ψ|EP×{1}, then g is fiber-preserving homotopic to f and g-1(EQ)=N.In 1974, Hatcher and Quinn [5] showed an interesting result as follows.Theorem 3 (Hatcher-Quinn).
Given smooth bundle mapsf:EP→EM and iQ:EQ→EM over a compact base space B which are immersions in each fiber, assume m>q+((p+k)/2)+1 and m>p+((q+k)/2)+1. Then there is a fiber-preserving map g:EP→EM over B homotopic to f such that g-1(EQ)=N.Their theorem required a mapf to be an immersion on each fiber which cannot be applied to our main theorem. We decided to give an alternative technique to proof the main theorem by constructing the required homotopy step by step. This techniques could be used as a model to achieve the similar result for the equivariant setting (see [6]).


## 2. Proof of the Main Theorem

Theorem 4 (Whitney embedding theorem).
LetMm and Nn be smooth manifolds, and let f:M→N be a smooth map. If n>2m, then f is homotopic to an embedding g:M→N.Lemma 5.
Letf:Mm→Nn be a map between two smooth manifolds. Let A be a closed submanifold of M. Assume that f|A is an embedding.
Ifn>2m, then f is homotopic to an embedding g relative to A.Proof.
LetT be a tubular neighborhood of A in M.
Step
1. Extend the embedding f|A:A→N to an embedding fT:T→N.
LetνA⊆M be the normal bundle of A in M and D(ν) denote the disc bundle of ν. Then the tubular neighborhood theorem implies D(νA⊆M)≅T.
Claim. For any given embedding A↪gN and vector bundle η over A, then
(9)(gextendstoanembeddingofD(η)intoN)⟺(ϕ:η→ν(g)thereexistsabundlemonomorphismϕ:η⟶ν(g)),
where ν(g) is the normal bundle of A in N via g.
Proof of Claim.
(
⇐
) We have a diagram
(10)
where exp is the exponential map.
Note thatexp(D(ν(g))≅ tubular neighborhood of A in N via g. Then exp∘ϕ:D(η)↪N is a desired embedding.
(
⇒
) Assume there exists an embedding gT so that the following diagram commutes
(11)
Thenν(g)≅η⊕i*ν(gT).
We are in the situation where we have a commutative diagram(12)
Letν(f):=f*τN-τM. Then i*(ν(f))⊕νA⊆M≅stableν(g), where ≅stable denotes the stable isomorphism between 2 vector bundles.
Ifn-a>a, then i*(ν(f))⊕νA⊆M≅ν(g), so there exists a bundle monomorphism
(13)
Apply the Claim wheng=f|A and η=νA⊆M. Then we have an extension embedding of g from D(νA⊆M)≅T↪fTN.
Step
2. We have a map f|M-intT:M∖int(T)→N and ∂(M∖int(T))=∂T.
The conditionb>(c+a/2+1) and Theorem 4 imply that f|M-int(T) is homotopic to an embedding gM-int(T).
Defineg:M→N by
(14)g(x)={gM-int(T)(x)ifx∈M∖int(T),gT(x)ifx∈T.
Thenf is homotopic to g relative to A.


### 2.1.  Proof of the Main Theorem

We divide the proof into3 steps.Step
1. Goal: fiber-preserving homotoped the map W→a:=πP∘𝒞EP to an embedding over B.By assumption, we have(15)W⟶𝒞E(f,iQ)⊆EP×EMI×EQ,
and we also have maps
(16)W→a:=πP∘𝒞EP,W→b:=πQ∘𝒞EQ,W×I⟶HEM,
where H:=K∘(𝒞×idI), so H|W×0=f∘a, H|W×1=b.Recall that∂W=f-1(EQ)⊔N, a|∂W is just the inclusion of f-1(EQ) and N into EP.Apply the conditionm>q+((p+k)/2)+1 to Lemma 5, there exists an embedding 𝒜≃a (rel ∂W). That is, we have a commutative diagram
(17)We have a mapW→b:=πQ∘𝒞EQ. By concatenating the homotopy H and f∘L together, we have a homotopy V:W×[1,2]→EM such that V(_,1)=𝒜 and V(_,2)=b. Hence, iQ∘b is homotopic to f∘𝒜.Next, we want to modify the homotopyV such that it is fiber preserving with respect to prM.Note that we have a commutative diagram(18)We can apply the homotopy lifting property forprQ to get a homotopy of b to b′ through V′ such that the following diagram commute:
(19)LetΨW:=b′:W→EQ. Then ΨW is a fiber-preserving map over B through the lifting V′.Step
2. Goal: construct a bundle isomorphism
(20)ν(𝒜)⊕ϵ1≅b′*(νEQ⊆EM),
where ϵ1 is the trivial bundle.SincedimW<rankν(𝒜), it is enough to give a stable equivalence between such bundles.Now, we have(21)W↪𝒜EP⊆SL⟹νW⊆SL≅ν(𝒜)⊕𝒜*(νEP⊆SL).We also have a commutative diagram(22)
(23)𝒜≃a=πP∘𝒞⟹𝒜*(νEP⊆SL)≅(πP∘𝒞)*(νEP⊆SL),b′≃b=πQ∘𝒞⟹b′*(νEQ⊆EM)≅(πQ∘𝒞)*(νEQ⊆EM).
Thus, the bundle map 𝒞^:νW⊆SL×I→ξ=πP*(νEP⊆SL)⊕πQ*(νEQ⊆EM) yields the following stable isomorphism
(24)νW⊆SL⊕ϵ1≅stable(πP∘𝒞)*(νEP⊆SL)⊕(πQ∘𝒞)*(νEQ⊆EM).
Putting (21), (23), and (24) together, we get
(25)ν(𝒜)⊕(πP∘𝒞)*(νEP⊆SL)⊕ϵ1≅stable(πP∘𝒞)*(νEP⊆SL)⊕b′*(νEQ⊆EM).
Consequently, we have
(26)ν(𝒜)⊕ϵ1≅b′*(νEQ⊆EM).
This implies that we did construct a bundle map
(27)
which gives us the extension of map b′ to the tubular neighborhood of W in EP. More precisely,
(28)ΨT:D(ν(𝒜))↪D(ν(𝒜)⊕ϵ1)⟶b′^D(νEQ⊆EM),
where D denotes the disc bundle. Note that ΨT⋔EQ and ΨT(∂D(η1))⊆EM∖EQ.Sinceν(𝒜)⊕ϵ1≅b′*(νEQ⊆EM), we can find a subbundle η2 of b′*(νEQ⊆EM) such that η2≅ν(𝒜). For simplicity, let η1:=ν(𝒜).Step
3. Goal: construct the fiber-preserving smooth map Ψ:EP×I→EM over B.Recall that we have(29)W↪D(ν(𝒜))≃D(ν(𝒜)⊕ϵ1)≅D(b′*(νEQ⊆EM)).Then there exists a neighborhoodD¯ of W in D(b′*(νEQ⊆EM)) such that D¯≃D(b′*(νEQ⊆EM)) and D¯≅D(ν(𝒜)).According to (19), we have a commutative diagram
(30)
where Ψ1(w,t)=(V′(w,t),t).Letη1=ν(𝒜⊕ϵ1),  η2=b*(νEQ⊆EM).According to (27), there exists a bundle η over W×I such that η|W×i=ηi for i=1,2.LetD1:=D(ν(𝒜)), D2:=D¯. Then Di↪D(ηi) is a homotopy equivalence for i=1,2 and also D1≅D2.SinceD1∪W×[1,2]∪D2↪D1×[1,2] is a cofibration and a homotopy equivalence, there exist an extension D1×[1,2]→Ψ^1EM×[1,2] such that the following diagram commutes
(31)Next we want to construct an embeddingW↪𝒜′D2×[2,3] such that the following hold:(i)
𝒜′(W)∩{D2×2}=f-1(EQ),(ii)
𝒜′(W)∩{D2×3}=N,(iii)
𝒜′⋔D2×∂[2,3].We start by lettingα:W→[2,3] be a smooth map such that α⋔∂[2,3], α-1(2)=f-1(EQ) and α-1(3)=N, and we also have an inclusion W↪iWD2.Let𝒜′:=iW×α. Then 𝒜′ is such a required map.By the construction, we have(32)D(ν(𝒜′))≅D2×[2,3].Letψ2 be the composition of the maps
(33)W↪𝒜′D2×[2,3]→ΨT|D2×id[2,3]M×[2,3]→projM.Define a mapΨ2:=ψ2×α:W→EM×[2,3].Using the fact thatD(ν(𝒜′))≅D2×[2,3], then D2×∂[2,3]∪𝒜′(W)↪D2×[2,3] is a cofibration and homotopy equivalence. Hence there exists an extension D2×[2,3]→Ψ^2EM×[2,3].Note that for(x,3)∈D2×3 such that Ψ^2(x,3)∈EQ×3, the map Ψ^2∣D2×3=ΨT forces that x has to be in W, so the definition of Ψ2 implies x∈N. Thus
(34)Ψ^2∣D2×3-1(EQ)=N.We define a mapΨ~:{EP×I}∪{D1×[1,2]}∪{D2×[2,3]}→EM×[0,3] by
(35)Ψ~(p,t)={(f(p),t)ift∈[0,1],(Ψ^1(p),t)ift∈[1,2],(Ψ^2(p),t)ift∈[2,3].ThenΨ~ is well-defined map over B by the construction.It’s not hard to see that{EP×I}∪{D1×[1,2]}∪{D2×[2,3]} is diffeomorphic to EP×I. Define the map Ψ to be the composition of maps
(36)EP×I⟶≅{EP×I}∪{D1×[1,2]}∪{D2×[2,3]}⟶Ψ~EM×[0,3]⟶projEM,
where proj is the projection to the first factor.Thus, we get a mapΨ:EP×I→EM over B so that Ψ|EP×0=f. By construction, Ψ⋔EQ and Ψ|EP×1-1(EQ)=N as required.Corollary 6.
Assumem>q+((p+k)/2)+1. Then the map f can be fiber-preserving homotoped to a map whose image does not intersect EQ if and only if [c,c^]=0∈Ωp+q+k-mfr(E(f,iQ);ξ).


## 2.1.  Proof of the Main Theorem

We divide the proof into3 steps.Step
1. Goal: fiber-preserving homotoped the map W→a:=πP∘𝒞EP to an embedding over B.By assumption, we have(15)W⟶𝒞E(f,iQ)⊆EP×EMI×EQ,
and we also have maps
(16)W→a:=πP∘𝒞EP,W→b:=πQ∘𝒞EQ,W×I⟶HEM,
where H:=K∘(𝒞×idI), so H|W×0=f∘a, H|W×1=b.Recall that∂W=f-1(EQ)⊔N, a|∂W is just the inclusion of f-1(EQ) and N into EP.Apply the conditionm>q+((p+k)/2)+1 to Lemma 5, there exists an embedding 𝒜≃a (rel ∂W). That is, we have a commutative diagram
(17)We have a mapW→b:=πQ∘𝒞EQ. By concatenating the homotopy H and f∘L together, we have a homotopy V:W×[1,2]→EM such that V(_,1)=𝒜 and V(_,2)=b. Hence, iQ∘b is homotopic to f∘𝒜.Next, we want to modify the homotopyV such that it is fiber preserving with respect to prM.Note that we have a commutative diagram(18)We can apply the homotopy lifting property forprQ to get a homotopy of b to b′ through V′ such that the following diagram commute:
(19)LetΨW:=b′:W→EQ. Then ΨW is a fiber-preserving map over B through the lifting V′.Step
2. Goal: construct a bundle isomorphism
(20)ν(𝒜)⊕ϵ1≅b′*(νEQ⊆EM),
where ϵ1 is the trivial bundle.SincedimW<rankν(𝒜), it is enough to give a stable equivalence between such bundles.Now, we have(21)W↪𝒜EP⊆SL⟹νW⊆SL≅ν(𝒜)⊕𝒜*(νEP⊆SL).We also have a commutative diagram(22)
(23)𝒜≃a=πP∘𝒞⟹𝒜*(νEP⊆SL)≅(πP∘𝒞)*(νEP⊆SL),b′≃b=πQ∘𝒞⟹b′*(νEQ⊆EM)≅(πQ∘𝒞)*(νEQ⊆EM).
Thus, the bundle map 𝒞^:νW⊆SL×I→ξ=πP*(νEP⊆SL)⊕πQ*(νEQ⊆EM) yields the following stable isomorphism
(24)νW⊆SL⊕ϵ1≅stable(πP∘𝒞)*(νEP⊆SL)⊕(πQ∘𝒞)*(νEQ⊆EM).
Putting (21), (23), and (24) together, we get
(25)ν(𝒜)⊕(πP∘𝒞)*(νEP⊆SL)⊕ϵ1≅stable(πP∘𝒞)*(νEP⊆SL)⊕b′*(νEQ⊆EM).
Consequently, we have
(26)ν(𝒜)⊕ϵ1≅b′*(νEQ⊆EM).
This implies that we did construct a bundle map
(27)
which gives us the extension of map b′ to the tubular neighborhood of W in EP. More precisely,
(28)ΨT:D(ν(𝒜))↪D(ν(𝒜)⊕ϵ1)⟶b′^D(νEQ⊆EM),
where D denotes the disc bundle. Note that ΨT⋔EQ and ΨT(∂D(η1))⊆EM∖EQ.Sinceν(𝒜)⊕ϵ1≅b′*(νEQ⊆EM), we can find a subbundle η2 of b′*(νEQ⊆EM) such that η2≅ν(𝒜). For simplicity, let η1:=ν(𝒜).Step
3. Goal: construct the fiber-preserving smooth map Ψ:EP×I→EM over B.Recall that we have(29)W↪D(ν(𝒜))≃D(ν(𝒜)⊕ϵ1)≅D(b′*(νEQ⊆EM)).Then there exists a neighborhoodD¯ of W in D(b′*(νEQ⊆EM)) such that D¯≃D(b′*(νEQ⊆EM)) and D¯≅D(ν(𝒜)).According to (19), we have a commutative diagram
(30)
where Ψ1(w,t)=(V′(w,t),t).Letη1=ν(𝒜⊕ϵ1),  η2=b*(νEQ⊆EM).According to (27), there exists a bundle η over W×I such that η|W×i=ηi for i=1,2.LetD1:=D(ν(𝒜)), D2:=D¯. Then Di↪D(ηi) is a homotopy equivalence for i=1,2 and also D1≅D2.SinceD1∪W×[1,2]∪D2↪D1×[1,2] is a cofibration and a homotopy equivalence, there exist an extension D1×[1,2]→Ψ^1EM×[1,2] such that the following diagram commutes
(31)Next we want to construct an embeddingW↪𝒜′D2×[2,3] such that the following hold:(i)
𝒜′(W)∩{D2×2}=f-1(EQ),(ii)
𝒜′(W)∩{D2×3}=N,(iii)
𝒜′⋔D2×∂[2,3].We start by lettingα:W→[2,3] be a smooth map such that α⋔∂[2,3], α-1(2)=f-1(EQ) and α-1(3)=N, and we also have an inclusion W↪iWD2.Let𝒜′:=iW×α. Then 𝒜′ is such a required map.By the construction, we have(32)D(ν(𝒜′))≅D2×[2,3].Letψ2 be the composition of the maps
(33)W↪𝒜′D2×[2,3]→ΨT|D2×id[2,3]M×[2,3]→projM.Define a mapΨ2:=ψ2×α:W→EM×[2,3].Using the fact thatD(ν(𝒜′))≅D2×[2,3], then D2×∂[2,3]∪𝒜′(W)↪D2×[2,3] is a cofibration and homotopy equivalence. Hence there exists an extension D2×[2,3]→Ψ^2EM×[2,3].Note that for(x,3)∈D2×3 such that Ψ^2(x,3)∈EQ×3, the map Ψ^2∣D2×3=ΨT forces that x has to be in W, so the definition of Ψ2 implies x∈N. Thus
(34)Ψ^2∣D2×3-1(EQ)=N.We define a mapΨ~:{EP×I}∪{D1×[1,2]}∪{D2×[2,3]}→EM×[0,3] by
(35)Ψ~(p,t)={(f(p),t)ift∈[0,1],(Ψ^1(p),t)ift∈[1,2],(Ψ^2(p),t)ift∈[2,3].ThenΨ~ is well-defined map over B by the construction.It’s not hard to see that{EP×I}∪{D1×[1,2]}∪{D2×[2,3]} is diffeomorphic to EP×I. Define the map Ψ to be the composition of maps
(36)EP×I⟶≅{EP×I}∪{D1×[1,2]}∪{D2×[2,3]}⟶Ψ~EM×[0,3]⟶projEM,
where proj is the projection to the first factor.Thus, we get a mapΨ:EP×I→EM over B so that Ψ|EP×0=f. By construction, Ψ⋔EQ and Ψ|EP×1-1(EQ)=N as required.Corollary 6.
Assumem>q+((p+k)/2)+1. Then the map f can be fiber-preserving homotoped to a map whose image does not intersect EQ if and only if [c,c^]=0∈Ωp+q+k-mfr(E(f,iQ);ξ).


## 3. Application to Fixed Point Theory

Letp:Mm+k→Bk be a smooth fiber bundle with compact fibers and k>2. Assume that B is a closed manifold. Let f:M→M be a smooth map over B. That is, p∘f=p.The fixed point set off is
(37)Fix(f):={x∈M∣f(x)=x}.We have a homotopy pull-back diagram(38)
where (i)
ℒfM:={α∈MI|f(α(0))=α(1)},(ii)
ev0 and ev1 are the evaluation map at 0 and 1 respectively,(iii)
Δ:= the diagonal map, defined by x↦(x,x),(iv)
Δf:= the twisted diagonal map, defined by x↦(x,f(x)),(v)
M×BM:= the fiber bundle over B with fiber over b∈B, given by Fb×Fb where Fb is the fiber of p over b.Proposition 7.
There exists a homotopy fromf to f1 such that Δf1⋔Δ.The proof relies on the work of Koźniewski [4], relating to B-manifolds. Let B be a smooth manifold. A B-manifold is a manifold X together with a locally trivial submersion p:X→B. A B-map is a smooth fiber-preserving map.Lemma 8.
LetX and Y be B-manifolds, and let Z be a B-submanifold of Y. Let g:X→Y be a B-map. Then, there is a fiber-preserving smooth B-homotopy Ht:X→Y such that H0=g and H1⋔Z.Proof.
See [7] for the proof.We have a transversal (pull-back) square(39)
where i is the inclusion. Transversality yields that ν(i)≅i*(ν(Δ))≅i*(τM).Choose an embeddingM↪Sm+k for sufficiently large k. Then we have
(40)νFix(f1)⊆Sm+k≅ν(i)⊕i*(νM⊆Sm+k)≅i*(τM)⊕i*(νM⊆Sp+k)≅ϵ.
We denote this bundle isomorphism by νFix(f1)⊆Sm+k→g^ϵ. We also have a map Fix(f1)→gℒfM defined by x↦cx, where cx is the constant map at x. Thus Lbord(f):=[Fix(f1),g,g^] determines the element in Ωkfr(ℒfM;ϵ).Applying Theorem2, We obtain the following corollary.Corollary 9 (converse of fiberwise Lefschetz fixed point theorem).
Letf:Mm+k→Mm+k be a smooth bundle map over the closed manifold Bk. Assume that m≥k+3. Then, f is fiber homotopic to a fixed point free map if and only if Lbord(f)=0∈Ωkfr(ℒfM;ϵ).Corollary 10.
Letf:Mm+k→Mm+k be a smooth bundle map over the closed manifold Bk. Assume that m≥k+3. If there is [N,N→gℒfM,νN⊆Sm+k→g^ϵ]∈Ωkfr(ℒfM;ϵ)f where N is a finite subset of M such that Lbord(f)=[N,g,g^], then the map f can be fiber-preserving homotoped to a map h such that Fix(h)=N.



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