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 Β© 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 β†’ 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 β†’ 𝑀 be a smooth self-map of a compact smooth manifold𝑀.

If𝑓has no fixed point, then the Lefschetz number𝐿(𝑓) = 0, where𝐿(𝑓) = βˆ‘π‘˜β‰₯0(βˆ’1)π‘˜Tr(π‘“βˆ—|π»π‘˜(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

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

From now on, the notations π‘‹π‘Ž means the smooth manifold 𝑋of dimension π‘Žand 𝐼 means the unit interval

[0, 1]. If π‘Œπ‘ is a submanifold ofπ‘‹π‘Ž,𝑓 : 𝑋 β†’ 𝑍, andπœ‚ is

a bundle over𝑍, then]π‘ŒβŠ†π‘‹means the normal bundle ofπ‘Œin

𝑋andπ‘“βˆ—(πœ‚)is a pull-back bundle ofπœ‚along the map𝑓. Later we define β€œframed bordism with coefficient in a bundle” as follows. Let 𝑋 be a smooth manifold with a bundleπœ‰over it. DefineΞ©π‘“π‘Ÿβˆ—(𝑋; πœ‰)to be the bordism groups of manifolds mapping to𝑋, together with a stable isomorphism of the normal bundle with the pullback ofπœ‰. 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+π‘˜π‘ƒ ,πΈπ‘ž+π‘˜π‘„ , andπΈπ‘š+π‘˜π‘€ are smooth fiber bundles over a compact manifoldπ΅π‘˜. Let𝑓 : 𝐸𝑝+π‘˜π‘ƒ β†’ πΈπ‘š+π‘˜π‘€ 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 β‹” 𝐸𝑄in𝐸𝑀(see [4]). The homotopy pullback is

𝐸 (𝑓, 𝑖𝑄)

:= {(π‘₯, πœ†, 𝑦) ∈ 𝐸𝑃× 𝐸𝐼𝑀× 𝐸𝑄| πœ† (0) =𝑓 (π‘₯) , πœ† (1)=𝑦} .

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We have a diagram which commutes up to homotopy

f

iQ

EP EM

EQ

E(f, iQ)

πœ‹P

πœ‹Q

(3)

whereπœ‹π‘ƒandπœ‹π‘„are the trivial projections; that is, we have a

homotopy𝐸(𝑓, 𝑖𝑄) Γ— 𝐼󳨀→ 𝐸𝐾 𝑀defined by𝐾(π‘₯, πœ†, 𝑦, 𝑑) = πœ†(𝑑),

𝑑 ∈ 𝐼.

We also have a map𝑐 : π‘“βˆ’1(𝐸𝑄) β†’ 𝐸(𝑓, 𝑖𝑄)defined by

π‘₯ 󳨃→ (π‘₯, 𝑐𝑓(π‘₯), 𝑓(π‘₯))where𝑐𝑓(π‘₯) =constant path in𝐸𝑀at

𝑓(π‘₯).

Transversality yields a bundle map

EQ

EP

fβˆ’1(E𝑄)βŠ† Eπ‘„βŠ†EM

f|fβˆ’1(E𝑄)

fβˆ’1(EQ)

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Choose an embedding𝐸𝑝+π‘˜π‘ƒ βŠ† 𝑆𝐿, for sufficiently large

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

π‘“βˆ’1(𝐸

𝑄)󳨅→ 𝐸𝑖 𝑝+π‘˜π‘ƒ βŠ† 𝑆𝐿. So]π‘“βˆ’1(𝐸

𝑄)βŠ†π‘†πΏ β‰…]π‘“βˆ’1(𝐸𝑄)βŠ†πΈπ‘ƒβŠ• 𝑖

βˆ—] πΈπ‘ƒβŠ†π‘†πΏ. The commutative diagram

f|fβˆ’1(E𝑄)

EQ

EP

fβˆ’1(EQ)

i c

πœ‹Q

E(f, iQ)

πœ‹P

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yields a bundle map

fβˆ’1(EQ) c E(f, iQ)

Μ‚c πœ‹βˆ—(

Eπ‘„βŠ†E𝑀)βŠ• πœ‹βˆ—(Eπ‘ƒβŠ†S𝐿) := 𝜁

fβˆ’1(E 𝑄)βŠ†S𝐿

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Thus, (𝑐, ̂𝑐) determines an element [𝑐, ̂𝑐] ∈ Ξ©π‘“π‘Ÿπ‘+π‘ž+π‘˜βˆ’π‘š

(𝐸(𝑓, 𝑖𝑄); πœ‰).

(II) Suppose that(𝑁󳨀→ 𝐸(𝑓, 𝑖𝑐1 𝑄),]π‘βŠ†π‘†πΏ 󳨀→ πœ‰)̂𝑐1 is another representative of[𝑐, ̂𝑐], where𝑁𝑝+π‘ž+π‘˜βˆ’π‘š βŠ† 𝐸𝑝+π‘˜π‘ƒ βŠ† 𝑆𝐿. This

means we have a normal bordism(π‘Šσ³¨€β†’ 𝐸(𝑓, 𝑖C 𝑄),]π‘Šσ³¨€β†’ πœ‰)Μ‚C between(𝑐, ̂𝑐)and(𝑐1, ̂𝑐1); that is

(i)π‘Šπ‘+π‘ž+π‘˜βˆ’π‘š+1βŠ† (𝑆𝐿× 𝐼),

(ii)πœ•π‘Š βŠ† (𝑆𝐿× πœ•πΌ),

(iii)π‘Š β‹” (𝑆𝐿× πœ•πΌ),

(iv)π‘Š ∩ (𝑆𝐿× 0) = π‘“βˆ’1(𝐸𝑄)andπ‘Š ∩ (𝑆𝐿× 1) = 𝑁

such that

C|π‘“βˆ’1(𝐸

𝑄)= 𝑐 : 𝑓

βˆ’1(𝐸

𝑄) 󳨀 β†’ 𝐸 (𝑓, 𝑖𝑄) ,

C|𝑁= 𝑐1: 𝑁 󳨀 β†’ 𝐸 (𝑓, 𝑖𝑄) ,

Μ‚

C|]

π‘“βˆ’1(𝐸𝑄)βŠ†π‘†πΏ = ̂𝑐, CΜ‚|]π‘βŠ†π‘†πΏ = ̂𝑐1.

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Theorem 2 (main theorem). Assumeπ‘š > π‘ž + ((𝑝 + π‘˜)/2) + 1. Then, there exists a smooth fiber-preserving map over𝐡

Ξ¨ : 𝐸𝑃× 𝐼 󳨀 β†’ 𝐸𝑀 (8)

such thatΞ¨|𝐸𝑃×{0}= 𝑓,Ξ¨ β‹” 𝐸𝑄andΞ¨|πΈβˆ’1

𝑃×{1}(𝐸𝑄) = 𝑁.

Note that if we let𝑔 = Ξ¨|𝐸𝑃×{1}, then𝑔is fiber-preserving homotopic to𝑓andπ‘”βˆ’1(𝐸𝑄) = 𝑁.

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

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

𝐸𝑃 β†’ 𝐸𝑀and𝑖𝑄 : 𝐸𝑄 β†’ 𝐸𝑀over a compact base space

𝐡which are immersions in each fiber, assumeπ‘š > π‘ž + ((𝑝 +

π‘˜)/2) + 1andπ‘š > 𝑝 + ((π‘ž + π‘˜)/2) + 1. Then there is a

fiber-preserving map𝑔 : 𝐸𝑃 β†’ 𝐸𝑀over𝐡homotopic to𝑓such that

π‘”βˆ’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). Letπ‘€π‘šand𝑁𝑛 be smooth manifolds, and let𝑓 : 𝑀 β†’ 𝑁be a smooth map. If

𝑛 > 2π‘š, then𝑓is homotopic to an embedding𝑔 : 𝑀 β†’ 𝑁.

Lemma 5. Let𝑓 : π‘€π‘š β†’ 𝑁𝑛be a map between two smooth manifolds. Let𝐴be a closed submanifold of𝑀. Assume that

𝑓|𝐴is an embedding.

If𝑛 > 2π‘š, then𝑓is homotopic to an embedding𝑔relative

to𝐴.

Proof. Let𝑇be a tubular neighborhood of𝐴in𝑀.

Step 1. Extend the embedding𝑓|𝐴: 𝐴 β†’ 𝑁to an embedding

𝑓𝑇: 𝑇 β†’ 𝑁.

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

πœ‚over𝐴, then

(𝑔extends to an embedding of𝐷 (πœ‚) into𝑁)

⇐⇒ (there exists a bundle monomorphism

πœ™ : πœ‚ 󳨀 β†’](𝑔)) ,

(9)

where](𝑔)is the normal bundle of𝐴in𝑁via𝑔.

Proof of Claim.(⇐)We have a diagram

πœ™

N A

Zero section

β‰…

D((g))

exp D(πœ‚)

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where exp is the exponential map.

Note that exp(𝐷(](𝑔)) β‰…tubular neighborhood of𝐴in𝑁 via𝑔. Then exp∘ πœ™ : 𝐷(πœ‚) 󳨅→ 𝑁is a desired embedding.

(β‡’)Assume there exists an embedding𝑔𝑇 so that the following diagram commutes

D(πœ‚) gT

i g

A

N

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Then](𝑔) β‰… πœ‚ βŠ• π‘–βˆ—](𝑔𝑇).

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) := π‘“βˆ—πœπ‘βˆ’ πœπ‘€. Thenπ‘–βˆ—(](𝑓)) βŠ•]π΄βŠ†π‘€stableβ‰… ](𝑔),

wherestable≅ denotes the stable isomorphism between 2 vector bundles.

If𝑛 βˆ’ π‘Ž > π‘Ž, thenπ‘–βˆ—(](𝑓)) βŠ•]π΄βŠ†π‘€β‰…](𝑔), so there exists

a bundle monomorphism

A

(g)

AβŠ†M

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Apply the Claim when𝑔 = 𝑓|𝐴andπœ‚ = ]π΄βŠ†π‘€. Then we

have an extension embedding of𝑔from𝐷(]π΄βŠ†π‘€) β‰… 𝑇󳨅→ 𝑁𝑓𝑇 .

Step 2. We have a map𝑓|π‘€βˆ’int𝑇: 𝑀 \int(𝑇) β†’ 𝑁andπœ•(𝑀 \

int(𝑇)) = πœ•π‘‡.

The condition𝑏 > (𝑐 + π‘Ž/2 + 1)andTheorem 4imply that

𝑓|π‘€βˆ’int(𝑇)is homotopic to an embeddingπ‘”π‘€βˆ’int(𝑇). Define𝑔 : 𝑀 β†’ 𝑁by

𝑔 (π‘₯) = {π‘”π‘€βˆ’int(𝑇)(π‘₯) if π‘₯ ∈ 𝑀 \int(𝑇) ,

𝑔𝑇(π‘₯) if π‘₯ ∈ 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 mapπ‘Šσ³¨€σ³¨€σ³¨€σ³¨€σ³¨€σ³¨€β†’π‘Ž:=πœ‹π‘ƒβˆ˜C

𝐸𝑃to an embedding over𝐡. By assumption, we have

π‘Šσ³¨€ β†’ 𝐸 (𝑓, 𝑖C 𝑄) βŠ† 𝐸𝑃× 𝐸𝐼𝑀× 𝐸𝑄, (15)

and we also have maps

π‘Šσ³¨€σ³¨€σ³¨€σ³¨€σ³¨€σ³¨€β†’ πΈπ‘Ž:=πœ‹π‘ƒβˆ˜C

𝑃, π‘Š

𝑏:=πœ‹π‘„βˆ˜C

󳨀󳨀󳨀󳨀󳨀󳨀→ 𝐸𝑄, π‘Š Γ— 𝐼󳨀 β†’ 𝐸𝐻 𝑀,

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where𝐻 := 𝐾 ∘ (CΓ—id𝐼), so𝐻|π‘ŠΓ—0= 𝑓 ∘ π‘Ž, 𝐻|π‘ŠΓ—1= 𝑏. Recall thatπœ•π‘Š = π‘“βˆ’1(𝐸𝑄) βŠ” 𝑁,π‘Ž|πœ•π‘Šis just the inclusion ofπ‘“βˆ’1(𝐸𝑄)and𝑁into𝐸𝑃.

Apply the conditionπ‘š > π‘ž + ((𝑝 + π‘˜)/2) + 1toLemma 5, there exists an embeddingA≃ π‘Ž(relπœ•π‘Š). That is, we have a commutative diagram

L

EP

a π’œ W Γ— 1

W Γ— 0

W Γ— I (17)

We have a map π‘Š 󳨀󳨀󳨀󳨀󳨀󳨀→ 𝐸𝑏:=πœ‹π‘„βˆ˜C 𝑄. By concatenating the homotopy𝐻and 𝑓 ∘ 𝐿together, we have a homotopy 𝑉 :

π‘Š Γ— [1, 2] β†’ 𝐸𝑀such that𝑉( , 1) = Aand𝑉( , 2) = 𝑏.

Hence,π‘–π‘„βˆ˜ 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

Vσ³°€

EQ

B ∘V

prQ

prM W Γ— 2

W Γ—[1, 2]

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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

EQβŠ† EM

bσ³°€:= Vσ³°€

W Γ— 1

Vσ³°€

π’œ

f

iQ

prQ

prM

prP W Γ— 2

W Γ— 1

W Γ—[1, 2] (19)

LetΞ¨π‘Š:= 𝑏󸀠 : π‘Š β†’ 𝐸𝑄. ThenΞ¨π‘Šis a fiber-preserving

map over𝐡through the lifting𝑉󸀠.

Step 2. Goal: construct a bundle isomorphism

](A) βŠ• πœ–1β‰… π‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€) , (20)

whereπœ–1is the trivial bundle.

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

Now, we have

π‘Šσ³¨…β†’ 𝐸A 𝑃 βŠ† 𝑆𝐿󳨐⇒]π‘ŠβŠ†π‘†πΏ β‰…](A) βŠ•Aβˆ—(]𝐸

π‘ƒβŠ†π‘†πΏ) . (21)

We also have a commutative diagram

W

EQ

EP

bσ³°€

π’œ πœ‹P

πœ‹Q

E(f,iQ)

π’ž

(22)

A≃ π‘Ž = πœ‹π‘ƒβˆ˜C󳨐⇒Aβˆ—(]πΈπ‘ƒβŠ†π‘†πΏ) β‰… (πœ‹π‘ƒβˆ˜C)βˆ—(]𝐸

π‘ƒβŠ†π‘†πΏ) ,

𝑏󸀠≃ 𝑏 = πœ‹

π‘„βˆ˜C󳨐⇒ π‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€) β‰… (πœ‹π‘„βˆ˜C)

βˆ—(] πΈπ‘„βŠ†πΈπ‘€) .

(23)

Thus, the bundle mapCΜ‚ : ]π‘ŠβŠ†π‘†πΏΓ—πΌ β†’ πœ‰ = πœ‹βˆ—π‘ƒ(]𝐸 π‘ƒβŠ†π‘†πΏ) βŠ•

πœ‹βˆ—

𝑄(]πΈπ‘„βŠ†πΈπ‘€)yields the following stable isomorphism

]π‘ŠβŠ†π‘†πΏβŠ• πœ–1

stable

β‰… (πœ‹π‘ƒβˆ˜C)βˆ—(]πΈπ‘ƒβŠ†π‘†πΏ) βŠ• (πœ‹π‘„βˆ˜C)βˆ—(]𝐸

π‘„βŠ†πΈπ‘€) . (24)

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

](A) βŠ• (πœ‹π‘ƒβˆ˜C)βˆ—(]πΈπ‘ƒβŠ†π‘†πΏ) βŠ• πœ–

1stable

β‰… (πœ‹π‘ƒβˆ˜C)βˆ—(]πΈπ‘ƒβŠ†π‘†πΏ)

βŠ• π‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€) .

(25)

Consequently, we have

](A) βŠ• πœ–1β‰… π‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€) . (26)

This implies that we did construct a bundle map

(π’œ)βŠ• πœ–1 Μ‚bσ³°€

bσ³°€

W EQ

Eπ‘„βŠ†E𝑀

(27)

which gives us the extension of map 𝑏󸀠 to the tubular neighborhood ofπ‘Šin𝐸𝑃. More precisely,

Ψ𝑇: 𝐷 (](A)) 󳨅→ 𝐷 (](A) βŠ• πœ–1)󳨀 β†’ 𝐷 (𝑏̂󸀠 ]πΈπ‘„βŠ†πΈπ‘€) , (28)

where𝐷 denotes the disc bundle. Note thatΨ𝑇 β‹” 𝐸𝑄 and

Ψ𝑇(πœ•π·(πœ‚1)) βŠ† 𝐸𝑀\ 𝐸𝑄.

Since](A) βŠ• πœ–1 β‰… π‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€), we can find a subbundle

πœ‚2ofπ‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€)such thatπœ‚2β‰…](A). For simplicity, letπœ‚1:= ](A).

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

𝐸𝑃× 𝐼 β†’ 𝐸𝑀over𝐡.

Recall that we have

π‘Š 󳨅→ 𝐷 (](A)) ≃ 𝐷 (](A) βŠ• πœ–1) β‰… 𝐷 (π‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€)) .

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Then there exists a neighborhood 𝐷 of π‘Š in 𝐷(π‘σΈ€ βˆ—

(]πΈπ‘„βŠ†πΈπ‘€))such that𝐷 ≃ 𝐷(π‘σΈ€ βˆ—(]πΈπ‘„βŠ†πΈπ‘€))and𝐷 β‰… 𝐷(](A)). According to (19), we have a commutative diagram

EM

EM

EM

Vσ³°€

|

Vσ³°€

| Ξ¨1 W Γ— 2

W Γ— 1

W Γ—[1, 2]

Γ— 2

Γ— 1

Γ—[1, 2]

W Γ— 2

W Γ— 1

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whereΞ¨1(𝑀, 𝑑) = (𝑉󸀠(𝑀, 𝑑), 𝑑).

Letπœ‚1=](AβŠ• πœ–1), πœ‚2= π‘βˆ—(]πΈπ‘„βŠ†πΈπ‘€).

According to (27), there exists a bundleπœ‚overπ‘Š Γ— 𝐼such

thatπœ‚|π‘ŠΓ—π‘–= πœ‚π‘–for𝑖 = 1, 2.

Let𝐷1 := 𝐷(](A)),𝐷2 := 𝐷. Then𝐷𝑖 󳨅→ 𝐷(πœ‚π‘–)is a

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𝐷1Γ— [1, 2] Μ‚ Ξ¨1

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

commutes

D1

D1 D2

[1, 2]

EMβ‰… D(Eπ‘„βŠ†E𝑀)

Μ‚

Ξ¨1

f|D1

exp E

MΓ— 2

EMΓ— 1 EMΓ—

[1, 2]

Γ—

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Next we want to construct an embeddingπ‘Š A σΈ€ 

󳨅→ 𝐷2Γ—[2, 3]

such that the following hold:

(i)AσΈ€ (π‘Š) ∩ {𝐷2Γ— 2} = π‘“βˆ’1(𝐸𝑄),

(ii)AσΈ€ (π‘Š) ∩ {𝐷2Γ— 3} = 𝑁,

(iii)AσΈ€ β‹” 𝐷2Γ— πœ•[2, 3].

We start by letting𝛼 : π‘Š β†’ [2, 3]be a smooth map such

that𝛼 β‹” πœ•[2, 3],π›Όβˆ’1(2) = π‘“βˆ’1(𝐸𝑄)andπ›Όβˆ’1(3) = 𝑁, and we

also have an inclusionπ‘Šσ³¨…β†’ π·π‘–π‘Š 2.

LetAσΈ€ := π‘–π‘ŠΓ— 𝛼. ThenAσΈ€ is such a required map. By the construction, we have

𝐷 (](AσΈ€ )) β‰… 𝐷2Γ— [2, 3] . (32)

Letπœ“2be the composition of the maps

π‘Šσ³¨…β†’ 𝐷AσΈ€  2Γ— [2, 3]󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ 𝑀 Γ— [2, 3]Ψ𝑇|𝐷2Γ—id[2,3] 󳨀󳨀󳨀→ 𝑀.proj (33)

Define a mapΞ¨2:= πœ“2Γ— 𝛼 : π‘Š β†’ 𝐸𝑀× [2, 3].

Using the fact that𝐷(](AσΈ€ )) β‰… 𝐷2Γ— [2, 3], then𝐷2Γ—

πœ•[2, 3] βˆͺAσΈ€ (π‘Š) 󳨅→ 𝐷

2Γ— [2, 3]is a cofibration and homotopy

equivalence. Hence there exists an extension𝐷2Γ— [2, 3]󳨀󳨀→̂Ψ2

𝐸𝑀× [2, 3].

Note that for(π‘₯, 3) ∈ 𝐷2Γ— 3such thatΞ¨Μ‚2(π‘₯, 3) ∈ 𝐸𝑄× 3, the mapΜ‚Ξ¨2|𝐷2Γ—3 = Ψ𝑇forces thatπ‘₯has to be inπ‘Š, so the definition ofΞ¨2impliesπ‘₯ ∈ 𝑁. Thus

Μ‚

Ξ¨2|π·βˆ’12Γ—3(𝐸𝑄) = 𝑁. (34)

We define a mapΜƒΞ¨ : {𝐸𝑃×𝐼}βˆͺ{𝐷1Γ—[1, 2]}βˆͺ{𝐷2Γ—[2, 3]} β†’

𝐸𝑀× [0, 3]by

Μƒ

Ξ¨ (𝑝, 𝑑) ={{{{

{

(𝑓 (𝑝) , 𝑑) if𝑑 ∈ [0, 1] ,

(Μ‚Ξ¨1(𝑝) , 𝑑) if𝑑 ∈ [1, 2] ,

(Μ‚Ξ¨2(𝑝) , 𝑑) if𝑑 ∈ [2, 3] .

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ThenΞ¨Μƒis well-defined map over𝐡by the construction.

It’s not hard to see that{𝐸𝑃× 𝐼} βˆͺ {𝐷1Γ— [1, 2]} βˆͺ {𝐷2Γ—

[2, 3]}is diffeomorphic to𝐸𝑃× 𝐼. Define the mapΞ¨to be the

composition of maps

𝐸𝑃× 𝐼󳨀 β†’ {𝐸≅ 𝑃× 𝐼} βˆͺ {𝐷1Γ— [1, 2]} βˆͺ {𝐷2Γ— [2, 3]}

Μƒ Ξ¨

󳨀 β†’ 𝐸𝑀× [0, 3]󳨀 β†’ 𝐸proj 𝑀,

(36)

where proj is the projection to the first factor.

Thus, we get a mapΞ¨ : 𝐸𝑃× 𝐼 β†’ 𝐸𝑀over𝐡so that

Ξ¨|𝐸𝑃×0 = 𝑓. By construction,Ξ¨ β‹” 𝐸𝑄andΞ¨

βˆ’1

|𝐸𝑃×1(𝐸𝑄) = 𝑁as

required.

Corollary 6. Assumeπ‘š > π‘ž + ((𝑝 + π‘˜)/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 ∈

Ξ©π‘“π‘Ÿπ‘+π‘ž+π‘˜βˆ’π‘š(𝐸(𝑓, 𝑖𝑄); πœ‰).

3. Application to Fixed Point Theory

Let𝑝 : π‘€π‘š+π‘˜ β†’ π΅π‘˜be a smooth fiber bundle with compact

fibers andπ‘˜ > 2. Assume that𝐡is a closed manifold. Let𝑓 :

𝑀 β†’ 𝑀be a smooth map over𝐡. That is,𝑝 ∘ 𝑓 = 𝑝.

The fixed point set of𝑓is

Fix(𝑓) := {π‘₯ ∈ 𝑀 | 𝑓 (π‘₯) = π‘₯} . (37)

We have a homotopy pull-back diagram

β„’fM e0

e1

M

M

Ξ”

Ξ”f M Γ—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)Ξ” :=the diagonal map, defined byπ‘₯ 󳨃→ (π‘₯, π‘₯), (iv)Δ𝑓 := the twisted diagonal map, defined by π‘₯ 󳨃→

(π‘₯, 𝑓(π‘₯)),

(v)𝑀×𝐡𝑀 :=the fiber bundle over𝐡with fiber over𝑏 ∈

𝐡, given by𝐹𝑏× 𝐹𝑏where𝐹𝑏is the fiber of𝑝over𝑏.

Proposition 7. There exists a homotopy from𝑓to𝑓1such that

Δ𝑓1β‹” Ξ”.

The proof relies on the work of KoΒ΄zniewski [4], relating to𝐡-manifolds. Let𝐡be a smooth manifold. A𝐡-manifold is a manifold𝑋together with a locally trivial submersion𝑝 :

𝑋 β†’ 𝐡. A𝐡-map is a smooth fiber-preserving map.

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is a fiber-preserving smooth𝐡-homotopy𝐻𝑑 : 𝑋 β†’ π‘Œsuch that𝐻0= 𝑔and𝐻1β‹” 𝑍.

Proof. See [7] for the proof.

We have a transversal (pull-back) square

f1) i

i

M

M

Ξ”

M Γ— M

Ξ”f1

Fix(

(39)

where 𝑖 is the inclusion. Transversality yields that ](𝑖) β‰…

π‘–βˆ—(](Ξ”)) β‰… π‘–βˆ—(πœπ‘€).

Choose an embedding𝑀 󳨅→ π‘†π‘š+π‘˜for sufficiently largeπ‘˜. Then we have

]Fix(𝑓1)βŠ†π‘†π‘š+π‘˜ β‰…](𝑖) βŠ• π‘–βˆ—(]π‘€βŠ†π‘†π‘š+π‘˜)

β‰… π‘–βˆ—(πœπ‘€) βŠ• π‘–βˆ—(]π‘€βŠ†π‘†π‘+π‘˜) β‰… πœ–.

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We denote this bundle isomorphism by]Fix(𝑓1)βŠ†π‘†π‘š+π‘˜

̂𝑔

󳨀→ πœ–. We

also have a map Fix(𝑓1) 󳨀→𝑔 L𝑓𝑀defined byπ‘₯ 󳨃→ 𝑐π‘₯, where

𝑐π‘₯is the constant map atπ‘₯. Thus𝐿bord(𝑓) := [Fix(𝑓1), 𝑔, ̂𝑔] determines the element inΞ©π‘“π‘Ÿπ‘˜ (L𝑓𝑀; πœ–).

ApplyingTheorem 2, We obtain the following corollary.

Corollary 9 (converse of fiberwise Lefschetz fixed point theorem). Let𝑓 : π‘€π‘š+π‘˜ β†’ π‘€π‘š+π‘˜be a smooth bundle map over the closed manifoldπ΅π‘˜. Assume thatπ‘š β‰₯ π‘˜ + 3. Then,

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

𝐿bord(𝑓) = 0 ∈ Ξ©π‘“π‘Ÿ

π‘˜ (L𝑓𝑀; πœ–).

Corollary 10. Let𝑓 : π‘€π‘š+π‘˜ β†’ π‘€π‘š+π‘˜be a smooth bundle map over the closed manifoldπ΅π‘˜. Assume thatπ‘š β‰₯ π‘˜+3. If there

is[𝑁, 𝑁󳨀→𝑔 L𝑓𝑀,]π‘βŠ†π‘†π‘š+π‘˜ 󳨀→ πœ–] ∈ Ω̂𝑔 π‘“π‘Ÿπ‘˜ (L𝑓𝑀; πœ–)𝑓where𝑁

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

Fix(β„Ž) = 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, β€œFixpunktklassen. I,”Mathematische Annalen, vol.

117, pp. 659–671, 1941 (German).

[2] J. Nielsen, β€œ Β¨Uber die Minimalzahl der Fixpunkte bei den

Abbildungstypen der RingflΒ¨achen,” Mathematische Annalen,

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

[3] B. J. Jiang, β€œFixed 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. KoΒ΄zniewski, β€œThe category of submersions,”L’AcadΒ΄emie

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

[5] A. Hatcher and F. Quinn, β€œBordism 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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