A Bordism Viewpoint of Fiberwise Intersections

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Volume 2013, Article ID 430105,6pages http://dx.doi.org/10.1155/2013/430105

Research Article

A Bordism Viewpoint of Fiberwise Intersections

Gun Sunyeekhan

50 Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand

Correspondence should be addressed to Gun Sunyeekhan; fscigsy@ku.ac.th

Received 1 March 2013; Accepted 3 June 2013

Academic Editor: Nawab Hussain

Copyright © 2013 Gun Sunyeekhan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 manifold𝑀, when is𝑓homotopic 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 manifold𝑀.

If𝑓has no fixed point, then the Lefschetz number𝐿(𝑓) = 0, where𝐿(𝑓) = ∑_𝑘≥0(−1)𝑘Tr(𝑓_∗|𝐻_𝑘(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 space

𝐵. The proof of the main theorem depends heavily on the intersection problem as follows.

From now on, the notations 𝑋𝑎 means the smooth manifold 𝑋of dimension 𝑎and 𝐼 means the unit interval

[0, 1]. If 𝑌𝑏 is a submanifold of𝑋𝑎,𝑓 : 𝑋 → 𝑍, and𝜂 is

a bundle over𝑍, then]_𝑌⊆𝑋means the normal bundle of𝑌in

𝑋and𝑓∗(𝜂)is a pull-back bundle of𝜂along the map𝑓. Later we define “framed bordism with coefficient in a bundle” as follows. Let 𝑋 be a smooth manifold with a bundle𝜉over it. DefineΩ𝑓𝑟_∗(𝑋; 𝜉)to be the bordism groups of manifolds mapping to𝑋, together with a stable isomorphism of the normal bundle with the pullback of𝜉. This framed

bordism group will be a home for our invariants𝐿bord(𝑓) (described in the last section) which detects more fixed point information than the regular Lefschetz number.

(I) Suppose that 𝐸𝑝+𝑘_𝑃 ,𝐸𝑞+𝑘_𝑄 , and𝐸𝑚+𝑘_𝑀 are smooth fiber bundles over a compact manifold𝐵𝑘. Let𝑓 : 𝐸𝑝+𝑘_𝑃 → 𝐸𝑚+𝑘_𝑀 be a bundle map, and let𝐸_𝑄be a subbundle of𝐸_𝑀with the inclusion bundle map𝑖_𝑄: 𝐸_𝑄󳨅→ 𝐸_𝑀.

We have a commutative diagram

Pp Mm Qq

f iQ

E_P E_M E_Q

B prP

prM

prQ

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where 𝑃, 𝑄, and 𝑀 are the fibers of pr_𝑃, pr_𝑄, and pr_𝑀, respectively.

We may assume that𝑓 ⋔ 𝐸_𝑄in𝐸_𝑀(see [4]). The homotopy pullback is

𝐸 (𝑓, 𝑖𝑄)

:= {(𝑥, 𝜆, 𝑦) ∈ 𝐸𝑃× 𝐸𝐼𝑀× 𝐸𝑄| 𝜆 (0) =𝑓 (𝑥) , 𝜆 (1)=𝑦} .

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We have a diagram which commutes up to homotopy

f

iQ

E_P E_M

E_Q

E(f, i_Q)

𝜋P

𝜋Q

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where𝜋_𝑃and𝜋_𝑄are the trivial projections; that is, we have a

homotopy𝐸(𝑓, 𝑖_𝑄) × 𝐼󳨀→ 𝐸𝐾 _𝑀defined by𝐾(𝑥, 𝜆, 𝑦, 𝑡) = 𝜆(𝑡),

𝑡 ∈ 𝐼.

We also have a map𝑐 : 𝑓−1(𝐸_𝑄) → 𝐸(𝑓, 𝑖_𝑄)defined by

𝑥 󳨃→ (𝑥, 𝑐𝑓(𝑥), 𝑓(𝑥))where𝑐𝑓(𝑥) =constant path in𝐸𝑀at

𝑓(𝑥).

Transversality yields a bundle map

E_Q

EP

_f−1_(E𝑄)⊆ _E_𝑄_⊆_E_M

f_|f−1_(E𝑄)

f−1(E_Q)

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Choose an embedding𝐸𝑝+𝑘_𝑃 ⊆ 𝑆𝐿, for sufficiently large

𝐿, where 𝑆𝐿 is a sphere of dimension 𝐿. Then, we have

𝑓−1_(𝐸

𝑄)󳨅→ 𝐸𝑖 𝑝+𝑘𝑃 ⊆ 𝑆𝐿. So]𝑓−1_(𝐸

𝑄)⊆𝑆𝐿 ≅]𝑓−1(𝐸𝑄)⊆𝐸𝑃⊕ 𝑖

∗_] 𝐸𝑃⊆𝑆𝐿. The commutative diagram

f_|f−1_(E𝑄)

E_Q

E_P

f−1(E_Q)

i c

𝜋Q

E(f, i_Q)

𝜋P

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yields a bundle map

f−1(E_Q) c E(f, iQ)

̂c _𝜋∗₍

E𝑄⊆E𝑀)⊕ 𝜋∗(E𝑃⊆S𝐿) := 𝜁

_f−1_(E 𝑄)⊆S𝐿

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Thus, (𝑐, ̂𝑐) determines an element [𝑐, ̂𝑐] ∈ Ω𝑓𝑟_{𝑝+𝑞+𝑘−𝑚}

(𝐸(𝑓, 𝑖_𝑄); 𝜉).

(II) Suppose that(𝑁󳨀→ 𝐸(𝑓, 𝑖𝑐1 _𝑄),]_𝑁⊆𝑆𝐿 󳨀→ 𝜉)̂𝑐1 is another representative of[𝑐, ̂𝑐], where𝑁𝑝+𝑞+𝑘−𝑚 ⊆ 𝐸𝑝+𝑘_𝑃 ⊆ 𝑆𝐿. This

means we have a normal bordism(𝑊󳨀→ 𝐸(𝑓, 𝑖C _𝑄),]_𝑊󳨀→ 𝜉)̂C between(𝑐, ̂𝑐)and(𝑐₁, ̂𝑐₁); that is

(i)𝑊𝑝+𝑞+𝑘−𝑚+1⊆ (𝑆𝐿× 𝐼),

(ii)𝜕𝑊 ⊆ (𝑆𝐿× 𝜕𝐼),

(iii)𝑊 ⋔ (𝑆𝐿× 𝜕𝐼),

(iv)𝑊 ∩ (𝑆𝐿× 0) = 𝑓−1(𝐸_𝑄)and𝑊 ∩ (𝑆𝐿× 1) = 𝑁

such that

C_|𝑓−1_(𝐸

𝑄)= 𝑐 : 𝑓

−1_(𝐸

𝑄) 󳨀 → 𝐸 (𝑓, 𝑖𝑄) ,

C_|𝑁= 𝑐₁: 𝑁 󳨀 → 𝐸 (𝑓, 𝑖_𝑄) ,

̂

C_|_]

𝑓−1(𝐸𝑄)⊆𝑆𝐿 = ̂𝑐, Ĉ|]𝑁⊆𝑆𝐿 = ̂𝑐1.

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Theorem 2 (main theorem). Assume𝑚 > 𝑞 + ((𝑝 + 𝑘)/2) + 1. Then, there exists a smooth fiber-preserving map over𝐵

Ψ : 𝐸_𝑃× 𝐼 󳨀 → 𝐸_𝑀 (8)

such thatΨ_|𝐸_𝑃_×{0}= 𝑓,Ψ ⋔ 𝐸_𝑄andΨ_|𝐸−1

𝑃×{1}(𝐸𝑄) = 𝑁.

Note that if we let𝑔 = Ψ_|𝐸_𝑃_×{1}, then𝑔is fiber-preserving homotopic to𝑓and𝑔−1(𝐸_𝑄) = 𝑁.

In 1974, Hatcher and Quinn [5] showed an interesting result as follows.

Theorem 3 (Hatcher-Quinn). Given smooth bundle maps𝑓 :

𝐸_𝑃 → 𝐸_𝑀and𝑖_𝑄 : 𝐸_𝑄 → 𝐸_𝑀over a compact base space

𝐵which are immersions in each fiber, assume𝑚 > 𝑞 + ((𝑝 +

𝑘)/2) + 1and𝑚 > 𝑝 + ((𝑞 + 𝑘)/2) + 1. Then there is a

fiber-preserving map𝑔 : 𝐸_𝑃 → 𝐸_𝑀over𝐵homotopic to𝑓such that

𝑔−1_(𝐸

𝑄) = 𝑁.

Their theorem required a map𝑓to 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). Let𝑀𝑚and𝑁𝑛 be smooth manifolds, and let𝑓 : 𝑀 → 𝑁be a smooth map. If

𝑛 > 2𝑚, then𝑓is homotopic to an embedding𝑔 : 𝑀 → 𝑁.

Lemma 5. Let𝑓 : 𝑀𝑚 → 𝑁𝑛be a map between two smooth manifolds. Let𝐴be a closed submanifold of𝑀. Assume that

𝑓_|𝐴is an embedding.

If𝑛 > 2𝑚, then𝑓is homotopic to an embedding𝑔relative

to𝐴.

Proof. Let𝑇be a tubular neighborhood of𝐴in𝑀.

Step 1. Extend the embedding𝑓_|𝐴: 𝐴 → 𝑁to an embedding

𝑓_𝑇: 𝑇 → 𝑁.

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Claim.For any given embedding𝐴󳨅→ 𝑁𝑔 and vector bundle

𝜂over𝐴, then

(𝑔extends to an embedding of𝐷 (𝜂) into𝑁)

⇐⇒ (there exists a bundle monomorphism

𝜙 : 𝜂 󳨀 →](𝑔)) ,

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where](𝑔)is the normal bundle of𝐴in𝑁via𝑔.

Proof of Claim.(⇐)We have a diagram

𝜙

N A

Zero section

≅

D((g))

exp D(𝜂)

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where exp is the exponential map.

Note that exp(𝐷(](𝑔)) ≅tubular neighborhood of𝐴in𝑁 via𝑔. Then exp∘ 𝜙 : 𝐷(𝜂) 󳨅→ 𝑁is a desired embedding.

(⇒)Assume there exists an embedding𝑔_𝑇 so that the following diagram commutes

D(𝜂) gT

i _g

A

N

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Then](𝑔) ≅ 𝜂 ⊕ 𝑖∗](𝑔_𝑇).

We are in the situation where we have a commutative diagram

i

f A

M N

f|A= g

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Let](𝑓) := 𝑓∗𝜏_𝑁− 𝜏_𝑀. Then𝑖∗(](𝑓)) ⊕]_𝐴⊆𝑀stable≅ ](𝑔),

wherestable≅ denotes the stable isomorphism between 2 vector bundles.

If𝑛 − 𝑎 > 𝑎, then𝑖∗(](𝑓)) ⊕]_𝐴⊆𝑀≅](𝑔), so there exists

a bundle monomorphism

A

(g)

_A⊆M

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Apply the Claim when𝑔 = 𝑓_|𝐴and𝜂 = ]_𝐴⊆𝑀. Then we

have an extension embedding of𝑔from𝐷(]_𝐴⊆𝑀) ≅ 𝑇󳨅→ 𝑁𝑓𝑇 .

Step 2. We have a map𝑓_|𝑀−_int_𝑇: 𝑀 \int(𝑇) → 𝑁and𝜕(𝑀 \

int(𝑇)) = 𝜕𝑇.

The condition𝑏 > (𝑐 + 𝑎/2 + 1)andTheorem 4imply that

𝑓_|𝑀−_int_(𝑇)is homotopic to an embedding𝑔_𝑀−_int_(𝑇). Define𝑔 : 𝑀 → 𝑁by

𝑔 (𝑥) = {𝑔𝑀−int(𝑇)(𝑥) if 𝑥 ∈ 𝑀 \int(𝑇) ,

𝑔_𝑇(𝑥) if 𝑥 ∈ 𝑇. (14)

Then𝑓is homotopic to𝑔relative to𝐴.

2.1. Proof of the Main Theorem. We divide the proof into3 steps.

Step 1. Goal: fiber-preserving homotoped the map𝑊󳨀󳨀󳨀󳨀󳨀󳨀→𝑎:=𝜋𝑃∘C

𝐸_𝑃to an embedding over𝐵. By assumption, we have

𝑊󳨀 → 𝐸 (𝑓, 𝑖C _𝑄) ⊆ 𝐸_𝑃× 𝐸𝐼_𝑀× 𝐸_𝑄, (15)

and we also have maps

𝑊_{󳨀󳨀󳨀󳨀󳨀󳨀→ 𝐸}𝑎:=𝜋𝑃∘C

𝑃, 𝑊

𝑏:=𝜋𝑄∘C

󳨀󳨀󳨀󳨀󳨀󳨀→ 𝐸_𝑄, 𝑊 × 𝐼󳨀 → 𝐸𝐻 _𝑀,

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where𝐻 := 𝐾 ∘ (C×id_𝐼), so𝐻_|𝑊×0= 𝑓 ∘ 𝑎, 𝐻_|𝑊×1= 𝑏. Recall that𝜕𝑊 = 𝑓−1(𝐸_𝑄) ⊔ 𝑁,𝑎_|𝜕𝑊is just the inclusion of𝑓−1(𝐸_𝑄)and𝑁into𝐸_𝑃.

Apply the condition𝑚 > 𝑞 + ((𝑝 + 𝑘)/2) + 1toLemma 5, there exists an embeddingA≃ 𝑎(rel𝜕𝑊). That is, we have a commutative diagram

L

E_P

a 𝒜 W × 1

W × 0

W × I (17)

We have a map 𝑊 󳨀󳨀󳨀󳨀󳨀󳨀→ 𝐸𝑏:=𝜋𝑄∘C _𝑄. By concatenating the homotopy𝐻and 𝑓 ∘ 𝐿together, we have a homotopy 𝑉 :

𝑊 × [1, 2] → 𝐸_𝑀such that𝑉( , 1) = Aand𝑉( , 2) = 𝑏.

Hence,𝑖_𝑄∘ 𝑏is homotopic to𝑓 ∘A.

Next, we want to modify the homotopy𝑉such that it is fiber preserving with respect to pr_𝑀.

Note that we have a commutative diagram

b

V󳰀

E_Q

B ∘V

prQ

prM W × 2

W ×[1, 2]

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We can apply the homotopy lifting property for pr_𝑄 to get a homotopy of𝑏to𝑏󸀠through𝑉󸀠such that the following diagram commute:

E

B Q

E_P

E_Q⊆ E_M

b󳰀_{:= V}󳰀

W × 1

V󳰀

𝒜

f

iQ

prQ

prM

prP W × 2

W × 1

W ×[1, 2] (19)

LetΨ_𝑊:= 𝑏󸀠 : 𝑊 → 𝐸_𝑄. ThenΨ_𝑊is a fiber-preserving

map over𝐵through the lifting𝑉󸀠.

Step 2. Goal: construct a bundle isomorphism

](A) ⊕ 𝜖1≅ 𝑏󸀠∗(]𝐸𝑄⊆𝐸𝑀) , (20)

where𝜖1is the trivial bundle.

Since dim 𝑊 < rank](A), it is enough to give a stable equivalence between such bundles.

Now, we have

𝑊󳨅→ 𝐸A 𝑃 ⊆ 𝑆𝐿󳨐⇒]𝑊⊆𝑆𝐿 ≅](A) ⊕A∗(]_𝐸

𝑃⊆𝑆𝐿) . (21)

We also have a commutative diagram

W

E_Q

E_P

b󳰀

𝒜 𝜋P

𝜋Q

E(f,i_Q)

𝒞

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A≃ 𝑎 = 𝜋_𝑃∘C󳨐⇒A∗(]_𝐸_𝑃_⊆𝑆𝐿) ≅ (𝜋_𝑃∘C)∗(]_𝐸

𝑃⊆𝑆𝐿) ,

𝑏󸀠_{≃ 𝑏 = 𝜋}

𝑄∘C󳨐⇒ 𝑏󸀠∗(]𝐸𝑄⊆𝐸𝑀) ≅ (𝜋𝑄∘C)

∗₍_] 𝐸𝑄⊆𝐸𝑀) .

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Thus, the bundle mapĈ : ]_𝑊⊆𝑆𝐿_×𝐼 → 𝜉 = 𝜋∗_𝑃(]_𝐸 𝑃⊆𝑆𝐿) ⊕

𝜋∗

𝑄(]𝐸𝑄⊆𝐸𝑀)yields the following stable isomorphism

]_𝑊⊆𝑆𝐿⊕ 𝜖1

stable

≅ (𝜋_𝑃∘C)∗(]_𝐸_𝑃_⊆𝑆𝐿) ⊕ (𝜋_𝑄∘C)∗(]_𝐸

𝑄⊆𝐸𝑀) . (24)

Putting (21), (23), and (24) together, we get

](A) ⊕ (𝜋𝑃∘C)∗(]𝐸𝑃⊆𝑆𝐿) ⊕ 𝜖

1stable

≅ (𝜋_𝑃∘C)∗(]_𝐸_𝑃_⊆𝑆𝐿)

⊕ 𝑏󸀠∗(]_𝐸_𝑄_⊆𝐸_𝑀) .

(25)

Consequently, we have

](A) ⊕ 𝜖1≅ 𝑏󸀠∗(]_𝐸_𝑄_⊆𝐸_𝑀) . (26)

This implies that we did construct a bundle map

(𝒜)⊕ 𝜖1 ̂b󳰀

b󳰀

W E_Q

_E_𝑄_⊆E_𝑀

(27)

which gives us the extension of map 𝑏󸀠 to the tubular neighborhood of𝑊in𝐸_𝑃. More precisely,

Ψ_𝑇: 𝐷 (](A)) 󳨅→ 𝐷 (](A) ⊕ 𝜖1)󳨀 → 𝐷 (𝑏̂󸀠 ]_𝐸_𝑄_⊆𝐸_𝑀) , (28)

where𝐷 denotes the disc bundle. Note thatΨ_𝑇 ⋔ 𝐸_𝑄 and

Ψ_𝑇(𝜕𝐷(𝜂₁)) ⊆ 𝐸_𝑀\ 𝐸_𝑄.

Since](A) ⊕ 𝜖1 ≅ 𝑏󸀠∗(]_𝐸_𝑄_⊆𝐸_𝑀), we can find a subbundle

𝜂₂of𝑏󸀠∗(]_𝐸_𝑄_⊆𝐸_𝑀)such that𝜂₂≅](A). For simplicity, let𝜂₁:= ](A).

Step 3. Goal: construct the fiber-preserving smooth mapΨ :

𝐸_𝑃× 𝐼 → 𝐸_𝑀over𝐵.

Recall that we have

𝑊 󳨅→ 𝐷 (](A)) ≃ 𝐷 (](A) ⊕ 𝜖1) ≅ 𝐷 (𝑏󸀠∗(]_𝐸_𝑄_⊆𝐸_𝑀)) .

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Then there exists a neighborhood 𝐷 of 𝑊 in 𝐷(𝑏󸀠∗

(]_𝐸_𝑄_⊆𝐸_𝑀))such that𝐷 ≃ 𝐷(𝑏󸀠∗(]_𝐸_𝑄_⊆𝐸_𝑀))and𝐷 ≅ 𝐷(](A)). According to (19), we have a commutative diagram

E_M

V󳰀

|

V󳰀

| Ψ1 W × 2

W × 1

W ×[1, 2]

× 2

× 1

×[1, 2]

W × 2

W × 1

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whereΨ₁(𝑤, 𝑡) = (𝑉󸀠(𝑤, 𝑡), 𝑡).

Let𝜂₁=](A⊕ 𝜖1), 𝜂₂= 𝑏∗(]_𝐸_𝑄_⊆𝐸_𝑀).

According to (27), there exists a bundle𝜂over𝑊 × 𝐼such

that𝜂_|𝑊×𝑖= 𝜂_𝑖for𝑖 = 1, 2.

Let𝐷₁ := 𝐷(](A)),𝐷₂ := 𝐷. Then𝐷_𝑖 󳨅→ 𝐷(𝜂_𝑖)is a

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𝐷1× [1, 2] ̂ Ψ1

󳨀󳨀→ 𝐸𝑀× [1, 2]such that the following diagram

commutes

D₁

D₁ D₂

[1, 2]

E_M≅ D(_E_𝑄_⊆E_𝑀)

̂

Ψ1

f|D1

exp _E

M× 2

E_M× 1 E_M×

[1, 2]

×

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Next we want to construct an embedding𝑊 A 󸀠

󳨅→ 𝐷₂×[2, 3]

such that the following hold:

(i)A󸀠(𝑊) ∩ {𝐷₂× 2} = 𝑓−1(𝐸_𝑄),

(ii)A󸀠(𝑊) ∩ {𝐷₂× 3} = 𝑁,

(iii)A󸀠⋔ 𝐷₂× 𝜕[2, 3].

We start by letting𝛼 : 𝑊 → [2, 3]be a smooth map such

that𝛼 ⋔ 𝜕[2, 3],𝛼−1(2) = 𝑓−1(𝐸_𝑄)and𝛼−1(3) = 𝑁, and we

also have an inclusion𝑊󳨅→ 𝐷𝑖𝑊 ₂.

LetA󸀠:= 𝑖_𝑊× 𝛼. ThenA󸀠is such a required map. By the construction, we have

𝐷 (](A󸀠)) ≅ 𝐷2× [2, 3] . (32)

Let𝜓₂be the composition of the maps

𝑊󳨅→ 𝐷A󸀠 ₂× [2, 3]󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ 𝑀 × [2, 3]Ψ𝑇|𝐷2×id[2,3] 󳨀󳨀󳨀→ 𝑀.proj (33)

Define a mapΨ₂:= 𝜓₂× 𝛼 : 𝑊 → 𝐸_𝑀× [2, 3].

Using the fact that𝐷(](A󸀠)) ≅ 𝐷₂× [2, 3], then𝐷₂×

𝜕[2, 3] ∪A󸀠_{(𝑊) 󳨅→ 𝐷}

2× [2, 3]is a cofibration and homotopy

equivalence. Hence there exists an extension𝐷₂× [2, 3]󳨀󳨀→̂Ψ2

𝐸𝑀× [2, 3].

Note that for(𝑥, 3) ∈ 𝐷₂× 3such thatΨ̂₂(𝑥, 3) ∈ 𝐸_𝑄× 3, the map̂Ψ_2|𝐷₂_×3 = Ψ_𝑇forces that𝑥has to be in𝑊, so the definition ofΨ₂implies𝑥 ∈ 𝑁. Thus

̂

Ψ_2|𝐷−1₂_×3(𝐸𝑄) = 𝑁. (34)

We define a map̃Ψ : {𝐸_𝑃×𝐼}∪{𝐷₁×[1, 2]}∪{𝐷₂×[2, 3]} →

𝐸𝑀× [0, 3]by

̃

Ψ (𝑝, 𝑡) ={{_{_{

{

(𝑓 (𝑝) , 𝑡) if𝑡 ∈ [0, 1] ,

(̂Ψ₁(𝑝) , 𝑡) if𝑡 ∈ [1, 2] ,

(̂Ψ₂(𝑝) , 𝑡) if𝑡 ∈ [2, 3] .

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ThenΨ̃is well-defined map over𝐵by the construction.

It’s not hard to see that{𝐸_𝑃× 𝐼} ∪ {𝐷₁× [1, 2]} ∪ {𝐷₂×

[2, 3]}is diffeomorphic to𝐸_𝑃× 𝐼. Define the mapΨto be the

composition of maps

𝐸_𝑃× 𝐼󳨀 → {𝐸≅ _𝑃× 𝐼} ∪ {𝐷₁× [1, 2]} ∪ {𝐷2× [2, 3]}

̃ Ψ

󳨀 → 𝐸_𝑀× [0, 3]󳨀 → 𝐸proj _𝑀,

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where proj is the projection to the first factor.

Thus, we get a mapΨ : 𝐸_𝑃× 𝐼 → 𝐸_𝑀over𝐵so that

Ψ|𝐸𝑃×0 = 𝑓. By construction,Ψ ⋔ 𝐸𝑄andΨ

−1

|𝐸𝑃×1(𝐸𝑄) = 𝑁as

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 𝐸_𝑄 if and only if [𝑐, ̂𝑐] = 0 ∈

Ω𝑓𝑟_{𝑝+𝑞+𝑘−𝑚}(𝐸(𝑓, 𝑖𝑄); 𝜉).

3. Application to Fixed Point Theory

Let𝑝 : 𝑀𝑚+𝑘 → 𝐵𝑘be a smooth fiber bundle with compact

fibers and𝑘 > 2. Assume that𝐵is a closed manifold. Let𝑓 :

𝑀 → 𝑀be a smooth map over𝐵. That is,𝑝 ∘ 𝑓 = 𝑝.

The fixed point set of𝑓is

Fix(𝑓) := {𝑥 ∈ 𝑀 | 𝑓 (𝑥) = 𝑥} . (37)

We have a homotopy pull-back diagram

ℒ_fM e0

e1

M

Δ

Δf M ×BM

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where

(i)L_𝑓𝑀 := {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 by𝑥 󳨃→ (𝑥, 𝑥), (iv)Δ_𝑓 := the twisted diagonal map, defined by 𝑥 󳨃→

(𝑥, 𝑓(𝑥)),

(v)𝑀×_𝐵𝑀 :=the fiber bundle over𝐵with fiber over𝑏 ∈

𝐵, given by𝐹_𝑏× 𝐹_𝑏where𝐹_𝑏is the fiber of𝑝over𝑏.

Proposition 7. There exists a homotopy from𝑓to𝑓₁such that

Δ_𝑓₁⋔ Δ.

The proof relies on the work of Ko´zniewski [4], relating to𝐵-manifolds. Let𝐵be a smooth manifold. A𝐵-manifold is a manifold𝑋together with a locally trivial submersion𝑝 :

𝑋 → 𝐵. A𝐵-map is a smooth fiber-preserving map.

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is a fiber-preserving smooth𝐵-homotopy𝐻_𝑡 : 𝑋 → 𝑌such that𝐻₀= 𝑔and𝐻₁⋔ 𝑍.

Proof. See [7] for the proof.

We have a transversal (pull-back) square

f₁) i

i

M

Δ

M × M

Δf1

Fix(

(39)

where 𝑖 is the inclusion. Transversality yields that ](𝑖) ≅

𝑖∗₍_]_{(Δ)) ≅ 𝑖}∗_(𝜏𝑀)_.

Choose an embedding𝑀 󳨅→ 𝑆𝑚+𝑘for sufficiently large𝑘. Then we have

]_Fix_(𝑓₁_)⊆𝑆𝑚+𝑘 ≅](𝑖) ⊕ 𝑖∗(]_𝑀⊆𝑆𝑚+𝑘)

≅ 𝑖∗(𝜏𝑀) ⊕ 𝑖∗(]_𝑀⊆𝑆𝑝+𝑘) ≅ 𝜖.

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We denote this bundle isomorphism by]_Fix_(𝑓₁_)⊆𝑆𝑚+𝑘

̂𝑔

󳨀→ 𝜖. We

also have a map Fix(𝑓₁) 󳨀→𝑔 L_𝑓𝑀defined by𝑥 󳨃→ 𝑐_𝑥, where

𝑐_𝑥is the constant map at𝑥. Thus𝐿bord(𝑓) := [Fix(𝑓₁), 𝑔, ̂𝑔] determines the element inΩ𝑓𝑟_𝑘 (L_𝑓𝑀; 𝜖).

ApplyingTheorem 2, We obtain the following corollary.

Corollary 9 (converse of fiberwise Lefschetz fixed point theorem). Let𝑓 : 𝑀𝑚+𝑘 → 𝑀𝑚+𝑘be a smooth bundle map over the closed manifold𝐵𝑘. Assume that𝑚 ≥ 𝑘 + 3. Then,

𝑓is fiber homotopic to a fixed point free map if and only if

𝐿bord_{(𝑓) = 0 ∈ Ω}𝑓𝑟

𝑘 (L𝑓𝑀; 𝜖).

Corollary 10. Let𝑓 : 𝑀𝑚+𝑘 → 𝑀𝑚+𝑘be a smooth bundle map over the closed manifold𝐵𝑘. Assume that𝑚 ≥ 𝑘+3. If there

is[𝑁, 𝑁󳨀→𝑔 L_𝑓𝑀,]_𝑁⊆𝑆𝑚+𝑘 󳨀→ 𝜖] ∈ Ω̂𝑔 𝑓𝑟_𝑘 (L_𝑓𝑀; 𝜖)𝑓where𝑁

is a finite subset of𝑀such that𝐿bord(𝑓) = [𝑁, 𝑔, ̂𝑔], then the map𝑓can be fiber-preserving homotoped to a mapℎsuch that

Fix(ℎ) = 𝑁.

Acknowledgments

The author gratefully acknowledge the Coordinating Center for Thai Government Science and Technology Scholarship Students (CSTS) and the National Science and Technology Development Agency (NSTDA) for funding this research. The author would also like to thank the referees, P.M. Akhmet’ev Akhmet’ev and Hichem Ben-El-Mechaiekh, for all the comments and suggestions.

References

[1] F. Wecken, “Fixpunktklassen. I,”Mathematische Annalen, vol.

117, pp. 659–671, 1941 (German).

[2] J. Nielsen, “ ¨Uber die Minimalzahl der Fixpunkte bei den

Abbildungstypen der Ringfl¨achen,” Mathematische Annalen,

vol. 82, no. 1-2, pp. 83–93, 1920 (German).

[3] B. J. Jiang, “Fixed point classes from a differential viewpoint,” in

Fixed Point Theory (Sherbrooke, Que., 1980), vol. 886 ofLecture Notes in Mathematics, pp. 163–170, Springer, Berlin, Germany, 1981.

[4] T. Ko´zniewski, “The category of submersions,”L’Acad´emie

Pol-onaise des Sciences. Bulletin. S´erie des Sciences Math´ematiques, vol. 27, no. 3-4, pp. 321–326, 1979.

[5] A. Hatcher and F. Quinn, “Bordism invariants of intersections

of submanifolds,”Transactions of the American Mathematical

Society, vol. 200, pp. 327–344, 1974.

[6] G. Sunyeekhan,Equivariant intersection theory [Ph.D. thesis],

University of Notre Dame, Notre Dame, Indiana, 2010.

[7] V. Coufal,A family version of Lefschetz-Nielsen fixed point theory

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