# A Bordism Viewpoint of Fiberwise Intersections

## Full text

(1)

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

## 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

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∗|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 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+𝑘𝑃 ,𝐸𝑞+𝑘𝑄 , and𝐸𝑚+𝑘𝑀 are smooth fiber bundles over a compact manifold𝐵𝑘. Let𝑓 : 𝐸𝑝+𝑘𝑃 → 𝐸𝑚+𝑘𝑀 be a bundle map, and let𝐸𝑄be a subbundle of𝐸𝑀with the inclusion bundle map𝑖𝑄: 𝐸𝑄󳨅→ 𝐸𝑀.

We have a commutative diagram

Pp Mm Qq

f iQ

EP EM EQ

B prP

prM

prQ

(1)

where 𝑃, 𝑄, and 𝑀 are the fibers of pr𝑃, pr𝑄, and pr𝑀, respectively.

We may assume that𝑓 ⋔ 𝐸𝑄in𝐸𝑀(see ). The homotopy pullback is

𝐸 (𝑓, 𝑖𝑄)

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

(2)

We have a diagram which commutes up to homotopy

f

iQ

EP EM

EQ

E(f, iQ)

𝜋P

𝜋Q

(3)

where𝜋𝑃and𝜋𝑄are the trivial projections; that is, we have a

homotopy𝐸(𝑓, 𝑖𝑄) × 𝐼󳨀→ 𝐸𝐾 𝑀defined by𝐾(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))where𝑐𝑓(𝑥) =constant path in𝐸𝑀at

𝑓(𝑥).

Transversality yields a bundle map

EQ

EP

f−1(E𝑄)⊆ E𝑄EM

f|f−1(E𝑄)

f−1(EQ)

(4)

Choose an embedding𝐸𝑝+𝑘𝑃 ⊆ 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)

EQ

EP

f−1(EQ)

i c

𝜋Q

E(f, iQ)

𝜋P

(5)

yields a bundle map

f−1(EQ) c E(f, iQ)

̂c 𝜋(

E𝑄⊆E𝑀)⊕ 𝜋∗(E𝑃⊆S𝐿) := 𝜁

f−1(E 𝑄)⊆S𝐿

(6)

Thus, (𝑐, ̂𝑐) determines an element [𝑐, ̂𝑐] ∈ Ω𝑓𝑟𝑝+𝑞+𝑘−𝑚

(𝐸(𝑓, 𝑖𝑄); 𝜉).

(II) Suppose that(𝑁󳨀→ 𝐸(𝑓, 𝑖𝑐1 𝑄),]𝑁⊆𝑆𝐿 󳨀→ 𝜉)̂𝑐1 is another representative of[𝑐, ̂𝑐], where𝑁𝑝+𝑞+𝑘−𝑚 ⊆ 𝐸𝑝+𝑘𝑃 ⊆ 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(𝐸𝑄)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, 𝑖𝑄) ,

̂

C|]

𝑓−1(𝐸𝑄)⊆𝑆𝐿 = ̂𝑐, Ĉ|]𝑁⊆𝑆𝐿 = ̂𝑐1.

(7)

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

Ψ : 𝐸𝑃× 𝐼 󳨀 → 𝐸𝑀 (8)

such thatΨ|𝐸𝑃×{0}= 𝑓,Ψ ⋔ 𝐸𝑄andΨ|𝐸−1

𝑃×{1}(𝐸𝑄) = 𝑁.

Note that if we let𝑔 = Ψ|𝐸𝑃×{1}, then𝑔is fiber-preserving homotopic to𝑓and𝑔−1(𝐸𝑄) = 𝑁.

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

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

𝐸𝑃 → 𝐸𝑀and𝑖𝑄 : 𝐸𝑄 → 𝐸𝑀over 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 : 𝐸𝑃 → 𝐸𝑀over𝐵homotopic 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 ).

### 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

𝑓𝑇: 𝑇 → 𝑁.

(3)

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)) ,

(9)

where](𝑔)is the normal bundle of𝐴in𝑁via𝑔.

Proof of Claim.(⇐)We have a diagram

𝜙

N A

Zero section

D((g))

exp D(𝜂)

(10)

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 embedding𝑔𝑇 so that the following diagram commutes

D(𝜂) gT

i g

A

N

(11)

Then](𝑔) ≅ 𝜂 ⊕ 𝑖∗](𝑔𝑇).

We are in the situation where we have a commutative diagram

i

f A

M N

f|A= g

(12)

Let](𝑓) := 𝑓∗𝜏𝑁− 𝜏𝑀. 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

(13)

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−int𝑇: 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 embedding𝑔𝑀−int(𝑇). Define𝑔 : 𝑀 → 𝑁by

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

𝑔𝑇(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

𝐸𝑃to an embedding over𝐵. By assumption, we have

𝑊󳨀 → 𝐸 (𝑓, 𝑖C 𝑄) ⊆ 𝐸𝑃× 𝐸𝐼𝑀× 𝐸𝑄, (15)

and we also have maps

𝑊󳨀󳨀󳨀󳨀󳨀󳨀→ 𝐸𝑎:=𝜋𝑃∘C

𝑃, 𝑊

𝑏:=𝜋𝑄∘C

󳨀󳨀󳨀󳨀󳨀󳨀→ 𝐸𝑄, 𝑊 × 𝐼󳨀 → 𝐸𝐻 𝑀,

(16)

where𝐻 := 𝐾 ∘ (C×id𝐼), so𝐻|𝑊×0= 𝑓 ∘ 𝑎, 𝐻|𝑊×1= 𝑏. Recall that𝜕𝑊 = 𝑓−1(𝐸𝑄) ⊔ 𝑁,𝑎|𝜕𝑊is just the inclusion of𝑓−1(𝐸𝑄)and𝑁into𝐸𝑃.

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

L

EP

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] → 𝐸𝑀such that𝑉( , 1) = Aand𝑉( , 2) = 𝑏.

Hence,𝑖𝑄∘ 𝑏is homotopic to𝑓 ∘A.

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

Note that we have a commutative diagram

b

V󳰀

EQ

B ∘V

prQ

prM W × 2

W ×[1, 2]

(4)

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

E

B Q

EP

EQ⊆ EM

b󳰀:= V󳰀

W × 1

V󳰀

𝒜

f

iQ

prQ

prM

prP W × 2

W × 1

W ×[1, 2] (19)

LetΨ𝑊:= 𝑏󸀠 : 𝑊 → 𝐸𝑄. ThenΨ𝑊is 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

EQ

EP

b󳰀

𝒜 𝜋P

𝜋Q

E(f,iQ)

𝒞

(22)

A≃ 𝑎 = 𝜋𝑃∘C󳨐⇒A∗(]𝐸𝑃⊆𝑆𝐿) ≅ (𝜋𝑃∘C)∗(]𝐸

𝑃⊆𝑆𝐿) ,

𝑏󸀠≃ 𝑏 = 𝜋

𝑄∘C󳨐⇒ 𝑏󸀠∗(]𝐸𝑄⊆𝐸𝑀) ≅ (𝜋𝑄∘C)

(] 𝐸𝑄⊆𝐸𝑀) .

(23)

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

≅ (𝜋𝑃∘C)∗(]𝐸𝑃⊆𝑆𝐿) ⊕ (𝜋𝑄∘C)∗(]𝐸

𝑄⊆𝐸𝑀) . (24)

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

](A) ⊕ (𝜋𝑃∘C)∗(]𝐸𝑃⊆𝑆𝐿) ⊕ 𝜖

1stable

≅ (𝜋𝑃∘C)∗(]𝐸𝑃⊆𝑆𝐿)

⊕ 𝑏󸀠∗(]𝐸𝑄⊆𝐸𝑀) .

(25)

Consequently, we have

](A) ⊕ 𝜖1≅ 𝑏󸀠∗(]𝐸𝑄⊆𝐸𝑀) . (26)

This implies that we did construct a bundle map

(𝒜)⊕ 𝜖1 ̂b󳰀

b󳰀

W EQ

E𝑄⊆E𝑀

(27)

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

Ψ𝑇: 𝐷 (](A)) 󳨅→ 𝐷 (](A) ⊕ 𝜖1)󳨀 → 𝐷 (𝑏̂󸀠 ]𝐸𝑄⊆𝐸𝑀) , (28)

where𝐷 denotes the disc bundle. Note thatΨ𝑇 ⋔ 𝐸𝑄 and

Ψ𝑇(𝜕𝐷(𝜂1)) ⊆ 𝐸𝑀\ 𝐸𝑄.

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

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

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

𝐸𝑃× 𝐼 → 𝐸𝑀over𝐵.

Recall that we have

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

(29)

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

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

EM

EM

EM

V󳰀

|

V󳰀

| Ψ1 W × 2

W × 1

W ×[1, 2]

× 2

× 1

×[1, 2]

W × 2

W × 1

(30)

whereΨ1(𝑤, 𝑡) = (𝑉󸀠(𝑤, 𝑡), 𝑡).

Let𝜂1=](A⊕ 𝜖1), 𝜂2= 𝑏∗(]𝐸𝑄⊆𝐸𝑀).

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

that𝜂|𝑊×𝑖= 𝜂𝑖for𝑖 = 1, 2.

Let𝐷1 := 𝐷(](A)),𝐷2 := 𝐷. Then𝐷𝑖 󳨅→ 𝐷(𝜂𝑖)is a

(5)

𝐷1× [1, 2] ̂ Ψ1

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

commutes

D1

D1 D2

[1, 2]

EM≅ D(E𝑄⊆E𝑀)

̂

Ψ1

f|D1

exp E

M× 2

EM× 1 EM×

[1, 2]

×

(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

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

also have an inclusion𝑊󳨅→ 𝐷𝑖𝑊 2.

LetA󸀠:= 𝑖𝑊× 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 → 𝐸𝑀× [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) ∈ 𝐸𝑄× 3, the map̂Ψ2|𝐷2×3 = Ψ𝑇forces that𝑥has to be in𝑊, so the definition ofΨ2implies𝑥 ∈ 𝑁. Thus

̂

Ψ2|𝐷−12×3(𝐸𝑄) = 𝑁. (34)

We define a map̃Ψ : {𝐸𝑃×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] .

(35)

ThenΨ̃is well-defined map over𝐵by the construction.

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

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

composition of maps

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

̃ Ψ

󳨀 → 𝐸𝑀× [0, 3]󳨀 → 𝐸proj 𝑀,

(36)

where proj is the projection to the first factor.

Thus, we get a mapΨ : 𝐸𝑃× 𝐼 → 𝐸𝑀over𝐵so 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 𝐸𝑄 if and only if [𝑐, ̂𝑐] = 0 ∈

Ω𝑓𝑟𝑝+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

fM e0

e1

M

M

Δ

Δf M ×BM

(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 by𝑥 󳨃→ (𝑥, 𝑥), (iv)Δ𝑓 := the twisted diagonal map, defined by 𝑥 󳨃→

(𝑥, 𝑓(𝑥)),

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

𝐵, given by𝐹𝑏× 𝐹𝑏where𝐹𝑏is the fiber of𝑝over𝑏.

Proposition 7. There exists a homotopy from𝑓to𝑓1such that

Δ𝑓1⋔ Δ.

The proof relies on the work of Ko´zniewski , 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.

(6)

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

Proof. See  for the proof.

We have a transversal (pull-back) square

f1) i

i

M

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.

(40)

We denote this bundle isomorphism by]Fix(𝑓1)⊆𝑆𝑚+𝑘

̂𝑔

󳨀→ 𝜖. We

also have a map Fix(𝑓1) 󳨀→𝑔 L𝑓𝑀defined by𝑥 󳨃→ 𝑐𝑥, where

𝑐𝑥is the constant map at𝑥. Thus𝐿bord(𝑓) := [Fix(𝑓1), 𝑔, ̂𝑔] determines the element inΩ𝑓𝑟𝑘 (L𝑓𝑀; 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 ∈ Ω𝑓𝑟

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 L𝑓𝑀,]𝑁⊆𝑆𝑚+𝑘 󳨀→ 𝜖] ∈ Ω̂𝑔 𝑓𝑟𝑘 (L𝑓𝑀; 𝜖)𝑓where𝑁

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

 F. Wecken, “Fixpunktklassen. I,”Mathematische Annalen, vol.

117, pp. 659–671, 1941 (German).

 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).

 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.

 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.

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

of submanifolds,”Transactions of the American Mathematical

Society, vol. 200, pp. 327–344, 1974.

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

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

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

(7)

### http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Mathematics

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations International Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Complex Analysis

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Optimization

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Combinatorics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

### World Journal

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

## Discrete Mathematics

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Updating...