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 selfmap𝑓 : 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 selfmap 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⊆𝑋}means the normal bundle of𝑌in
𝑋and𝑓∗(𝜂)is a pullback 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_{𝑄}be a subbundle of𝐸_{𝑀}with 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
(1)
where 𝑃, 𝑄, and 𝑀 are the fibers of pr_{𝑃}, pr_{𝑄}, and pr_{𝑀}, respectively.
We may assume that𝑓 ⋔ 𝐸_{𝑄}in𝐸_{𝑀}(see [4]). The homotopy pullback is
𝐸 (𝑓, 𝑖𝑄)
:= {(𝑥, 𝜆, 𝑦) ∈ 𝐸𝑃× 𝐸𝐼𝑀× 𝐸𝑄 𝜆 (0) =𝑓 (𝑥) , 𝜆 (1)=𝑦} .
We have a diagram which commutes up to homotopy
f
iQ
E_{P} E_{M}
E_{Q}
E(f, i_{Q})
𝜋P
𝜋Q
(3)
where𝜋_{𝑃}and𝜋_{𝑄}are the trivial projections; that is, we have a
homotopy𝐸(𝑓, 𝑖_{𝑄}) × 𝐼→ 𝐸𝐾 _{𝑀}defined 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})
(4)
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
(5)
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
(6)
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⊆𝑆}𝐿 → 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(𝑐_{1}, ̂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(𝐸_{𝑄})and𝑊 ∩ (𝑆𝐿× 1) = 𝑁
such that
C_{𝑓}−1_{(𝐸}
𝑄)= 𝑐 : 𝑓
−1_{(𝐸}
𝑄) → 𝐸 (𝑓, 𝑖𝑄) ,
C_{𝑁}= 𝑐_{1}: 𝑁 → 𝐸 (𝑓, 𝑖_{𝑄}) ,
̂
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 fiberpreserving map over𝐵
Ψ : 𝐸_{𝑃}× 𝐼 → 𝐸_{𝑀} (8)
such thatΨ_{𝐸}_{𝑃}_{×{0}}= 𝑓,Ψ ⋔ 𝐸_{𝑄}andΨ_{𝐸}−1
𝑃×{1}(𝐸𝑄) = 𝑁.
Note that if we let𝑔 = Ψ_{𝐸}_{𝑃}_{×{1}}, then𝑔is fiberpreserving homotopic to𝑓and𝑔−1(𝐸_{𝑄}) = 𝑁.
In 1974, Hatcher and Quinn [5] showed an interesting result as follows.
Theorem 3 (HatcherQuinn). Given smooth bundle maps𝑓 :
𝐸_{𝑃} → 𝐸_{𝑀}and𝑖_{𝑄} : 𝐸_{𝑄} → 𝐸_{𝑀}over 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
fiberpreserving map𝑔 : 𝐸_{𝑃} → 𝐸_{𝑀}over𝐵homotopic 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_{𝐴}is 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.
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 embedding𝑔_{𝑇} so that the following diagram commutes
D(𝜂) gT
i _{g}
A
N
(11)
Then](𝑔) ≅ 𝜂 ⊕ 𝑖∗](𝑔_{𝑇}).
We are in the situation where we have a commutative diagram
i
f A
M N
fA= g
(12)
Let](𝑓) := 𝑓∗𝜏_{𝑁}− 𝜏_{𝑀}. Then𝑖∗(](𝑓)) ⊕]_{𝐴⊆𝑀}stable≅ ](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_{𝐴}and𝜂 = ]_{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: fiberpreserving homotoped the map𝑊→𝑎:=𝜋𝑃∘C
𝐸_{𝑃}to 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},
(16)
where𝐻 := 𝐾 ∘ (C×id_{𝐼}), so𝐻_{𝑊×0}= 𝑓 ∘ 𝑎, 𝐻_{𝑊×1}= 𝑏. Recall that𝜕𝑊 = 𝑓−1(𝐸_{𝑄}) ⊔ 𝑁,𝑎_{𝜕𝑊}is 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_{𝑀}such 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]
We can apply the homotopy lifting property for pr_{𝑄} to get a homotopy of𝑏to𝑏through𝑉such 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_{𝑊}is a fiberpreserving
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
(22)
A≃ 𝑎 = 𝜋_{𝑃}∘C⇒A∗(]_{𝐸}_{𝑃}_{⊆𝑆}𝐿) ≅ (𝜋_{𝑃}∘C)∗(]_{𝐸}
𝑃⊆𝑆𝐿) ,
𝑏_{≃ 𝑏 = 𝜋}
𝑄∘C⇒ 𝑏∗(]𝐸𝑄⊆𝐸𝑀) ≅ (𝜋𝑄∘C)
∗_{(}_{]} 𝐸𝑄⊆𝐸𝑀) .
(23)
Thus, the bundle mapĈ : ]_{𝑊⊆𝑆}𝐿_{×𝐼} → 𝜉 = 𝜋∗_{𝑃}(]_{𝐸} 𝑃⊆𝑆𝐿) ⊕
𝜋∗
𝑄(]𝐸𝑄⊆𝐸𝑀)yields the following stable isomorphism
]_{𝑊⊆𝑆}𝐿⊕ 𝜖1
stable
≅ (𝜋_{𝑃}∘C)∗(]_{𝐸}_{𝑃}_{⊆𝑆}𝐿) ⊕ (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}_{⊆𝑆}𝐿)
⊕ 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(𝜂_{1})) ⊆ 𝐸_{𝑀}\ 𝐸_{𝑄}.
Since](A) ⊕ 𝜖1 ≅ 𝑏∗(]_{𝐸}_{𝑄}_{⊆𝐸}_{𝑀}), we can find a subbundle
𝜂_{2}of𝑏∗(]_{𝐸}_{𝑄}_{⊆𝐸}_{𝑀})such that𝜂_{2}≅](A). For simplicity, let𝜂_{1}:= ](A).
Step 3. Goal: construct the fiberpreserving smooth mapΨ :
𝐸_{𝑃}× 𝐼 → 𝐸_{𝑀}over𝐵.
Recall that we have
𝑊 → 𝐷 (](A)) ≃ 𝐷 (](A) ⊕ 𝜖1) ≅ 𝐷 (𝑏∗(]_{𝐸}_{𝑄}_{⊆𝐸}_{𝑀})) .
(29)
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}
E_{M}
E_{M}
V

V
 Ψ1 W × 2
W × 1
W ×[1, 2]
× 2
× 1
×[1, 2]
W × 2
W × 1
(30)
whereΨ_{1}(𝑤, 𝑡) = (𝑉(𝑤, 𝑡), 𝑡).
Let𝜂_{1}=](A⊕ 𝜖1), 𝜂_{2}= 𝑏∗(]_{𝐸}_{𝑄}_{⊆𝐸}_{𝑀}).
According to (27), there exists a bundle𝜂over𝑊 × 𝐼such
that𝜂_{𝑊×𝑖}= 𝜂_{𝑖}for𝑖 = 1, 2.
Let𝐷_{1} := 𝐷(](A)),𝐷_{2} := 𝐷. Then𝐷_{𝑖} → 𝐷(𝜂_{𝑖})is a
𝐷1× [1, 2] ̂ Ψ1
→ 𝐸𝑀× [1, 2]such that the following diagram
commutes
D_{1}
D_{1} D_{2}
[1, 2]
E_{M}≅ D(_{E}_{𝑄}_{⊆E}_{𝑀})
̂
Ψ1
fD1
exp _{E}
M× 2
E_{M}× 1 E_{M}×
[1, 2]
×
(31)
Next we want to construct an embedding𝑊 A
→ 𝐷_{2}×[2, 3]
such that the following hold:
(i)A(𝑊) ∩ {𝐷_{2}× 2} = 𝑓−1(𝐸_{𝑄}),
(ii)A(𝑊) ∩ {𝐷_{2}× 3} = 𝑁,
(iii)A⋔ 𝐷_{2}× 𝜕[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. ThenAis such a required map. By the construction, we have
𝐷 (](A)) ≅ 𝐷2× [2, 3] . (32)
Let𝜓_{2}be the composition of the maps
𝑊→ 𝐷A _{2}× [2, 3]→ 𝑀 × [2, 3]Ψ𝑇𝐷2×id[2,3] → 𝑀.proj (33)
Define a mapΨ_{2}:= 𝜓_{2}× 𝛼 : 𝑊 → 𝐸_{𝑀}× [2, 3].
Using the fact that𝐷(](A)) ≅ 𝐷_{2}× [2, 3], then𝐷_{2}×
𝜕[2, 3] ∪A_{(𝑊) → 𝐷}
2× [2, 3]is a cofibration and homotopy
equivalence. Hence there exists an extension𝐷_{2}× [2, 3]→̂Ψ2
𝐸𝑀× [2, 3].
Note that for(𝑥, 3) ∈ 𝐷_{2}× 3such thatΨ̂_{2}(𝑥, 3) ∈ 𝐸_{𝑄}× 3, the map̂Ψ_{2𝐷}_{2}_{×3} = Ψ_{𝑇}forces that𝑥has to be in𝑊, so the definition ofΨ_{2}implies𝑥 ∈ 𝑁. Thus
̂
Ψ_{2𝐷}−1_{2}_{×3}(𝐸𝑄) = 𝑁. (34)
We define a map̃Ψ : {𝐸_{𝑃}×𝐼}∪{𝐷_{1}×[1, 2]}∪{𝐷_{2}×[2, 3]} →
𝐸𝑀× [0, 3]by
̃
Ψ (𝑝, 𝑡) ={{_{{}_{{}
{
(𝑓 (𝑝) , 𝑡) if𝑡 ∈ [0, 1] ,
(̂Ψ_{1}(𝑝) , 𝑡) if𝑡 ∈ [1, 2] ,
(̂Ψ_{2}(𝑝) , 𝑡) if𝑡 ∈ [2, 3] .
(35)
ThenΨ̃is welldefined map over𝐵by the construction.
It’s not hard to see that{𝐸_{𝑃}× 𝐼} ∪ {𝐷_{1}× [1, 2]} ∪ {𝐷_{2}×
[2, 3]}is diffeomorphic to𝐸_{𝑃}× 𝐼. Define the mapΨto be the
composition of maps
𝐸_{𝑃}× 𝐼 → {𝐸≅ _{𝑃}× 𝐼} ∪ {𝐷_{1}× [1, 2]} ∪ {𝐷2× [2, 3]}
̃ Ψ
→ 𝐸_{𝑀}× [0, 3] → 𝐸proj _{𝑀},
(36)
where proj is the projection to the first factor.
Thus, we get a mapΨ : 𝐸_{𝑃}× 𝐼 → 𝐸_{𝑀}over𝐵so that
Ψ𝐸𝑃×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 fiberpreserving 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 pullback diagram
ℒ_{f}M e0
e1
M
M
Δ
Δf M ×BM
(38)
where
(i)L_{𝑓}𝑀 := {𝛼 ∈ 𝑀𝐼 𝑓(𝛼(0)) = 𝛼(1)},
(ii)𝑒V_{0} and 𝑒V_{1} 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×_{𝐵}𝑀 :=the fiber bundle over𝐵with fiber over𝑏 ∈
𝐵, given by𝐹_{𝑏}× 𝐹_{𝑏}where𝐹_{𝑏}is the fiber of𝑝over𝑏.
Proposition 7. There exists a homotopy from𝑓to𝑓_{1}such that
Δ_{𝑓}_{1}⋔ Δ.
The proof relies on the work of Ko´zniewski [4], relating to𝐵manifolds. Let𝐵be a smooth manifold. A𝐵manifold is a manifold𝑋together with a locally trivial submersion𝑝 :
𝑋 → 𝐵. A𝐵map is a smooth fiberpreserving map.
is a fiberpreserving smooth𝐵homotopy𝐻_{𝑡} : 𝑋 → 𝑌such that𝐻_{0}= 𝑔and𝐻_{1}⋔ 𝑍.
Proof. See [7] for the proof.
We have a transversal (pullback) square
f_{1}) 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}_{(𝑓}_{1}_{)⊆𝑆}𝑚+𝑘 ≅](𝑖) ⊕ 𝑖∗(]_{𝑀⊆𝑆}𝑚+𝑘)
≅ 𝑖∗(𝜏𝑀) ⊕ 𝑖∗(]_{𝑀⊆𝑆}𝑝+𝑘) ≅ 𝜖.
(40)
We denote this bundle isomorphism by]_{Fix}_{(𝑓}_{1}_{)⊆𝑆}𝑚+𝑘
̂𝑔
→ 𝜖. We
also have a map Fix(𝑓_{1}) →𝑔 L_{𝑓}𝑀defined by𝑥 → 𝑐_{𝑥}, where
𝑐_{𝑥}is the constant map at𝑥. Thus𝐿bord(𝑓) := [Fix(𝑓_{1}), 𝑔, ̂𝑔] determines the element inΩ𝑓𝑟_{𝑘} (L_{𝑓}𝑀; 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_{𝑓}𝑀,]_{𝑁⊆𝑆}𝑚+𝑘 → 𝜖] ∈ Ω̂𝑔 𝑓𝑟_{𝑘} (L_{𝑓}𝑀; 𝜖)𝑓where𝑁
is a finite subset of𝑀such that𝐿bord(𝑓) = [𝑁, 𝑔, ̂𝑔], then the map𝑓can be fiberpreserving 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 BenElMechaiekh, 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. 12, 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
Polonaise des Sciences. Bulletin. S´erie des Sciences Math´ematiques, vol. 27, no. 34, 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 LefschetzNielsen fixed point theory
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