A Bordism Viewpoint of Fiberwise Intersections

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.

1. Introduction

In topological fixed point theory, there is a classical questions that people may ask; given a smooth self-map 𝑓∶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). Let 𝑓∶𝑀→𝑀 be a smooth self-map of a compact smooth manifold 𝑀.
If 𝑓 has no fixed point, then the Lefschetz number 𝐿(𝑓)=0, where 𝐿(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 𝑋^𝑎 means the smooth manifold 𝑋 of dimension 𝑎 and 𝐼 means the unit interval [0,1]. If 𝑌^𝑏 is a submanifold of 𝑋^𝑎, 𝑓∶𝑋→𝑍, and 𝜂 is a bundle over 𝑍, then 𝜈_𝑌⊆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. Let 𝑋 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 𝐿^bord(𝑓) (described in the last section) which detects more fixed point information than the regular Lefschetz number.

(I) Suppose that 𝐸_𝑃^𝑝+𝑘, 𝐸_𝑄^𝑞+𝑘, and 𝐸_𝑀^𝑚+𝑘 are smooth fiber bundles over a compact manifold 𝐵^𝑘. Let 𝑓∶𝐸_𝑃^𝑝+𝑘→𝐸_𝑀^𝑚+𝑘 be a bundle map, and let 𝐸_𝑄 be a subbundle of 𝐸_𝑀 with the inclusion bundle map 𝑖_𝑄∶𝐸_𝑄↪𝐸_𝑀.

We have a commutative diagram (1) where 𝑃, 𝑄, and 𝑀 are the fibers of pr_𝑃, pr_𝑄, and pr_𝑀, respectively.

 We may assume that 𝑓⋔𝐸_𝑄 in 𝐸_𝑀 (see [4]).

 The homotopy pullback is 𝐸𝑓,𝑖_𝑄∶=𝑥,𝜆,𝑦∈𝐸_𝑃×𝐸^𝐼_𝑀×𝐸_𝑄∣𝜆(0)=𝑓(𝑥),𝜆(1)=𝑦. (2)

We have a diagram which commutes up to homotopy (3) where 𝜋_𝑃 and 𝜋_𝑄 are the trivial projections; that is, we have a homotopy 𝐸(𝑓,𝑖_𝑄)×𝐼−→^𝐾𝐸_𝑀 defined by 𝐾(𝑥,𝜆,𝑦,𝑡)=𝜆(𝑡), 𝑡∈𝐼.

We also have a map 𝑐∶𝑓⁻¹(𝐸_𝑄)→𝐸(𝑓,𝑖_𝑄) defined by 𝑥↦(𝑥,𝑐_𝑓(𝑥),𝑓(𝑥)) where 𝑐_𝑓(𝑥)= constant path in 𝐸_𝑀 at 𝑓(𝑥).

Transversality yields a bundle map (4)

Choose an embedding 𝐸_𝑃^𝑝+𝑘⊆𝑆^𝐿, for sufficiently large 𝐿, where 𝑆^𝐿 is a sphere of dimension 𝐿. Then, we have 𝑓⁻¹(𝐸_𝑄)↪^𝑖𝐸_𝑃^𝑝+𝑘⊆𝑆^𝐿. So 𝜈_𝑓⁻¹_{(𝐸𝑄)⊆𝑆}^𝐿≅𝜈_𝑓⁻¹_{(𝐸𝑄)⊆𝐸𝑃}⊕𝑖^∗𝜈_{𝐸𝑃⊆𝑆}^𝐿.

The commutative diagram (5) yields a bundle map (6)

Thus, (𝑐,𝑐)̂ determines an element [𝑐,𝑐]̂∈Ω^𝑓𝑟_{𝑝+𝑞+𝑘−𝑚}(𝐸(𝑓,𝑖_𝑄);𝜉).

(II) Suppose that (𝑁−→^𝑐1𝐸(𝑓,𝑖_𝑄),𝜈_𝑁⊆𝑆^𝐿𝑐₁̂−−→𝜉) is another representative of [𝑐,𝑐]̂, where 𝑁^{𝑝+𝑞+𝑘−𝑚}⊆𝐸_𝑃^𝑝+𝑘⊆𝑆^𝐿. This means we have a normal bordism (𝑊−→^𝒞𝐸(𝑓,𝑖_𝑄), 𝜈_𝑊𝒞−−→𝜉) between (𝑐,𝑐)̂ and (𝑐₁,𝑐₁̂); that is (i)𝑊^{𝑝+𝑞+𝑘−𝑚+1}⊆(𝑆^𝐿×𝐼), (ii)𝜕𝑊⊆(𝑆^𝐿×𝜕𝐼), (iii)𝑊⋔(𝑆^𝐿×𝜕𝐼), (iv)𝑊∩(𝑆^𝐿×0)=𝑓⁻¹(𝐸_𝑄) and 𝑊∩(𝑆^𝐿×1)=𝑁such that 𝒞_|𝑓⁻¹_(𝐸𝑄)=𝑐∶𝑓⁻¹𝐸_𝑄⟶𝐸𝑓,𝑖_𝑄,𝒞_|𝑁=𝑐₁∶𝑁⟶𝐸𝑓,𝑖_𝑄,𝒞_{|𝜈𝑓𝑄𝐿−1(𝐸)⊆𝑆}=𝑐,̂𝒞_{|𝜈𝐿𝑁⊆𝑆}=𝑐₁̂. (7)

Theorem 2 (main theorem). Assume 𝑚>𝑞+((𝑝+𝑘)/2)+1. Then, there exists a smooth fiber-preserving map over 𝐵Ψ∶𝐸_𝑃×𝐼⟶𝐸_𝑀 (8) such that Ψ_{|𝐸𝑃×{0}}=𝑓, Ψ⋔𝐸_𝑄 and Ψ⁻¹_{|𝐸𝑃×{1}}(𝐸_𝑄)=𝑁.

Note that if we let 𝑔=Ψ_{|𝐸𝑃×{1}}, then 𝑔 is fiber-preserving homotopic to 𝑓 and 𝑔⁻¹(𝐸_𝑄)=𝑁.

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

Theorem 3 (Hatcher-Quinn). Given smooth bundle maps 𝑓∶𝐸_𝑃→𝐸_𝑀 and 𝑖_𝑄∶𝐸_𝑄→𝐸_𝑀 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∶𝐸_𝑃→𝐸_𝑀 over 𝐵 homotopic to 𝑓 such that 𝑔⁻¹(𝐸_𝑄)=𝑁.

Their theorem required a map 𝑓 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). Let 𝑀^𝑚 and 𝑁^𝑛 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. Let 𝑓∶𝑀^𝑚→𝑁^𝑛 be a map between two smooth manifolds. Let 𝐴 be a closed submanifold of 𝑀. Assume that 𝑓_|𝐴 is an embedding.
If 𝑛>2𝑚, then 𝑓 is homotopic to an embedding 𝑔 relative to 𝐴.

Proof. Let 𝑇 be a tubular neighborhood of 𝐴 in 𝑀.
Step 1. Extend the embedding 𝑓_|𝐴∶𝐴→𝑁 to an embedding 𝑓_𝑇∶𝑇→𝑁.
Let 𝜈_𝐴⊆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(𝜈_𝐴⊆𝑀)≅𝑇.
Claim. For any given embedding 𝐴↪^𝑔𝑁 and vector bundle 𝜂 over 𝐴, then 𝜂⟺𝑔,𝑔extendstoanembeddingof𝐷into𝑁thereexistsabundlemonomorphism𝜙∶𝜂⟶𝜈 (9) 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 that exp(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 𝑔_𝑇 so that the following diagram commutes (11)
Then 𝜈(𝑔)≅𝜂⊕𝑖^∗𝜈(𝑔_𝑇).
We are in the situation where we have a commutative diagram (12)
Let 𝜈(𝑓)∶=𝑓^∗𝜏_𝑁−𝜏_𝑀. Then 𝑖^∗(𝜈(𝑓))⊕𝜈≅_𝐴⊆𝑀^stable𝜈(𝑔), where ^st≅^able denotes the stable isomorphism between 2 vector bundles.
If 𝑛−𝑎>𝑎, then 𝑖^∗(𝜈(𝑓))⊕𝜈_𝐴⊆𝑀≅𝜈(𝑔), so there exists a bundle monomorphism (13)
Apply the Claim when 𝑔=𝑓_|𝐴 and 𝜂=𝜈_𝐴⊆𝑀. Then we have an extension embedding of 𝑔 from 𝐷(𝜈_𝐴⊆𝑀)≅𝑇↪^𝑓𝑇𝑁.
Step 2. We have a map 𝑓_{|𝑀−int𝑇}∶𝑀⧵int(𝑇)→𝑁 and 𝜕(𝑀⧵int(𝑇))=𝜕𝑇.
The condition 𝑏>(𝑐+𝑎/2+1) and Theorem 4 imply that 𝑓_{|𝑀−int(𝑇)} is homotopic to an embedding 𝑔_{𝑀−int(𝑇)}.
Define 𝑔∶𝑀→𝑁 by 𝑔(𝑥)=𝑔_{𝑀−int(𝑇)}(𝑥)(𝑇),𝑔if𝑥∈𝑀⧵int_𝑇(𝑥)if𝑥∈𝑇. (14)
Then 𝑓 is homotopic to 𝑔 relative to 𝐴.

2.1. Proof of the Main Theorem

We divide the proof into 3 steps.

Step 1. Goal: fiber-preserving homotoped the map 𝑊−−−−−−−→^{𝑎∶=𝜋𝑃∘𝒞}𝐸_𝑃 to an embedding over 𝐵.

By assumption, we have 𝑊⟶^𝒞𝐸𝑓,𝑖_𝑄⊆𝐸_𝑃×𝐸^𝐼_𝑀×𝐸_𝑄, (15) and we also have maps 𝑊−−−−−−−→^{𝑎∶=𝜋𝑃∘𝒞}𝐸_𝑃,𝑊−−−−−−−→^{𝑏∶=𝜋𝑄∘𝒞}𝐸_𝑄,𝑊×𝐼⟶^𝐻𝐸_𝑀, (16) where 𝐻∶=𝐾∘(𝒞×id_𝐼), so 𝐻_|𝑊×0=𝑓∘𝑎, 𝐻_|𝑊×1=𝑏.

Recall that 𝜕𝑊=𝑓⁻¹(𝐸_𝑄)⊔𝑁, 𝑎_|𝜕𝑊 is just the inclusion of 𝑓⁻¹(𝐸_𝑄) and 𝑁 into 𝐸_𝑃.

Apply the condition 𝑚>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 map 𝑊−−−−−−−→^{𝑏∶=𝜋𝑄∘𝒞}𝐸_𝑄. By concatenating the homotopy 𝐻 and 𝑓∘𝐿 together, we have a homotopy 𝑉∶𝑊×[1,2]→𝐸_𝑀 such that 𝑉(_,1)=𝒜 and 𝑉(_,2)=𝑏. Hence, 𝑖_𝑄∘𝑏 is homotopic to 𝑓∘𝒜.

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

Note that we have a commutative diagram (18)

We can apply the homotopy lifting property for pr_𝑄 to get a homotopy of 𝑏 to 𝑏^′ through 𝑉^′ such that the following diagram commute: (19)

Let Ψ_𝑊∶=𝑏^′∶𝑊→𝐸_𝑄. Then Ψ_𝑊 is a fiber-preserving map over 𝐵 through the lifting 𝑉^′.

Step 2. Goal: construct a bundle isomorphism 𝜈(𝒜)⊕𝜖¹≅𝑏^′∗𝜈_{𝐸𝑄⊆𝐸𝑀}, (20) where 𝜖¹ is the trivial bundle.

Since dim𝑊<rank𝜈(𝒜), it is enough to give a stable equivalence between such bundles.

Now, we have 𝑊↪^𝒜𝐸_𝑃⊆𝑆^𝐿⟹𝜈_𝑊⊆𝑆^𝐿≅𝜈(𝒜)⊕𝒜^∗𝜈_{𝐸𝑃⊆𝑆}^𝐿. (21)

We also have a commutative diagram (22)  𝒜≃𝑎=𝜋_𝑃∘𝒞⟹𝒜^∗𝜈_{𝐸𝑃⊆𝑆}^𝐿≅𝜋_𝑃∘𝒞^∗𝜈_{𝐸𝑃⊆𝑆}^𝐿,𝑏^′≃𝑏=𝜋_𝑄∘𝒞⟹𝑏^′∗𝜈_{𝐸𝑄⊆𝐸𝑀}≅𝜋_𝑄∘𝒞^∗𝜈_{𝐸𝑄⊆𝐸𝑀}. (23) Thus, the bundle map 𝒞∶𝜈_𝑊⊆𝑆^𝐿_×𝐼→𝜉=𝜋^∗_𝑃(𝜈_{𝐸𝑃⊆𝑆}^𝐿)⊕𝜋^∗_𝑄(𝜈_{𝐸𝑄⊆𝐸𝑀}) yields the following stable isomorphism 𝜈_𝑊⊆𝑆^𝐿⊕𝜖≅^1stable𝜋_𝑃∘𝒞^∗𝜈_{𝐸𝑃⊆𝑆}^𝐿⊕𝜋_𝑄∘𝒞^∗𝜈_{𝐸𝑄⊆𝐸𝑀}. (24) Putting (21), (23), and (24) together, we get 𝜈(𝒜)⊕𝜋_𝑃∘𝒞^∗𝜈_{𝐸𝑃⊆𝑆}^𝐿⊕𝜖≅^1stable𝜋_𝑃∘𝒞^∗𝜈_{𝐸𝑃⊆𝑆}^𝐿⊕𝑏^′∗𝜈_{𝐸𝑄⊆𝐸𝑀}. (25) Consequently, we have 𝜈(𝒜)⊕𝜖¹≅𝑏^′∗𝜈_{𝐸𝑄⊆𝐸𝑀}. (26) 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 𝐸_𝑃. More precisely, Ψ_𝑇∶𝐷(𝜈(𝒜))↪𝐷𝜈(𝒜)⊕𝜖¹𝑏^′⟶𝐷𝜈_{𝐸𝑄⊆𝐸𝑀}, (28) where 𝐷 denotes the disc bundle. Note that Ψ_𝑇⋔𝐸_𝑄 and Ψ_𝑇(𝜕𝐷(𝜂₁))⊆𝐸_𝑀⧵𝐸_𝑄.

Since 𝜈(𝒜)⊕𝜖¹≅𝑏^′∗(𝜈_{𝐸𝑄⊆𝐸𝑀}), we can find a subbundle 𝜂₂ of 𝑏^′∗(𝜈_{𝐸𝑄⊆𝐸𝑀}) such that 𝜂₂≅𝜈(𝒜). For simplicity, let 𝜂₁∶=𝜈(𝒜).

Step 3. Goal: construct the fiber-preserving smooth map Ψ∶𝐸_𝑃×𝐼→𝐸_𝑀 over 𝐵.

Recall that we have 𝑊↪𝐷(𝜈(𝒜))≃𝐷𝜈(𝒜)⊕𝜖¹≅𝐷𝑏^′∗𝜈_{𝐸𝑄⊆𝐸𝑀}. (29)

Then there exists a neighborhood 𝐷 of 𝑊 in 𝐷(𝑏^′∗(𝜈_{𝐸𝑄⊆𝐸𝑀})) such that 𝐷≃𝐷(𝑏^′∗(𝜈_{𝐸𝑄⊆𝐸𝑀})) and 𝐷≅𝐷(𝜈(𝒜)).

According to (19), we have a commutative diagram (30) where Ψ₁(𝑤,𝑡)=(𝑉^′(𝑤,𝑡),𝑡).

Let 𝜂₁=𝜈(𝒜⊕𝜖¹),  𝜂₂=𝑏^∗(𝜈_{𝐸𝑄⊆𝐸𝑀}).

According to (27), there exists a bundle 𝜂 over 𝑊×𝐼 such that 𝜂_|𝑊×𝑖=𝜂_𝑖 for 𝑖=1,2.

Let 𝐷₁∶=𝐷(𝜈(𝒜)), 𝐷₂∶=𝐷. Then 𝐷_𝑖↪𝐷(𝜂_𝑖) is a homotopy equivalence for 𝑖=1,2 and also 𝐷₁≅𝐷₂.

Since 𝐷₁∪𝑊×[1,2]∪𝐷₂↪𝐷₁×[1,2] is a cofibration and a homotopy equivalence, there exist an extension 𝐷₁Ψ×[1,2]−−−→¹𝐸_𝑀×[1,2] such that the following diagram commutes (31)

Next we want to construct an embedding 𝑊↪^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 letting 𝛼∶𝑊→[2,3] be a smooth map such that 𝛼⋔𝜕[2,3], 𝛼⁻¹(2)=𝑓⁻¹(𝐸_𝑄) and 𝛼⁻¹(3)=𝑁, and we also have an inclusion 𝑊↪^𝑖𝑊𝐷₂.

Let 𝒜^′∶=𝑖_𝑊×𝛼. Then 𝒜^′ is such a required map.

By the construction, we have 𝐷𝜈𝒜^′≅𝐷₂×[2,3]. (32)

Let 𝜓₂ be the composition of the maps 𝑊↪^𝒜′𝐷₂×[2,3]−−−−−−−−−−→^{Ψ𝑇2|𝐷×id[2,3]}𝑀×[2,3]−−−→^proj𝑀. (33)

Define a map Ψ₂∶=𝜓₂×𝛼∶𝑊→𝐸_𝑀×[2,3].

Using the fact that 𝐷(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 𝐷₂Ψ×[2,3]−−−→²𝐸_𝑀×[2,3].

Note that for (𝑥,3)∈𝐷₂×3 such that Ψ₂(𝑥,3)∈𝐸_𝑄×3, the map Ψ_2∣𝐷2×3=Ψ_𝑇 forces that 𝑥 has to be in 𝑊, so the definition of Ψ₂ implies 𝑥∈𝑁. Thus Ψ⁻¹_2∣𝐷2×3𝐸_𝑄=𝑁. (34)

We define a map Ψ∶{𝐸_𝑃×𝐼}∪{𝐷₁×[1,2]}∪{𝐷₂×[2,3]}→𝐸_𝑀×[0,3] by Ψ=𝑓𝑝[],𝑝,𝑡Ψ,𝑡if𝑡∈0,1₁𝑝[],Ψ,𝑡if𝑡∈1,2₂𝑝,𝑡if𝑡∈[2,3]. (35)

Then Ψ is well-defined map over 𝐵 by the construction.

It’s not hard to see that {𝐸_𝑃×𝐼}∪{𝐷₁×[1,2]}∪{𝐷₂×[2,3]} is diffeomorphic to 𝐸_𝑃×𝐼. Define the map Ψ to be the composition of maps 𝐸_𝑃×𝐼⟶^≅𝐸_𝑃×𝐼∪𝐷₁×[1,2]∪𝐷₂×[Ψ2,3]⟶𝐸_𝑀×[0,3]⟶^proj𝐸_𝑀, (36) where proj is the projection to the first factor.

Thus, we get a map Ψ∶𝐸_𝑃×𝐼→𝐸_𝑀 over 𝐵 so that Ψ_|𝐸𝑃×0=𝑓. By construction, Ψ⋔𝐸_𝑄 and Ψ⁻¹_|𝐸𝑃×1(𝐸_𝑄)=𝑁 as required.

Corollary 6. Assume 𝑚>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∈Ω^𝑓𝑟_{𝑝+𝑞+𝑘−𝑚}(𝐸(𝑓,𝑖_𝑄);𝜉).

3. Application to Fixed Point Theory

Let 𝑝∶𝑀^𝑚+𝑘→𝐵^𝑘 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 of 𝑓 is Fix𝑓∶=𝑥∈𝑀∣𝑓(𝑥)=𝑥. (37)

We have a homotopy pull-back diagram (38) where (i)ℒ_𝑓𝑀∶={𝛼∈𝑀^𝐼|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)Δ_𝑓∶= 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 𝐹_𝑏×𝐹_𝑏 where 𝐹_𝑏 is the fiber of 𝑝 over 𝑏.

Proposition 7. There exists a homotopy from 𝑓 to 𝑓₁ such that Δ_𝑓1⋔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. Let 𝑋 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 𝐻_𝑡∶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 embedding 𝑀↪𝑆^𝑚+𝑘 for sufficiently large 𝑘. Then we have 𝜈_{Fix(𝑓1)⊆𝑆}^𝑚+𝑘≅𝜈(𝑖)⊕𝑖^∗𝜈_𝑀⊆𝑆^𝑚+𝑘≅𝑖^∗(𝜏𝑀)⊕𝑖^∗𝜈_𝑀⊆𝑆^𝑝+𝑘≅𝜖. (40) We denote this bundle isomorphism by 𝜈_{Fix(𝑓1)⊆𝑆}^𝑚+𝑘𝑔−→𝜖. We also have a map Fix(𝑓₁)→−^𝑔ℒ_𝑓𝑀 defined by 𝑥↦𝑐_𝑥, where 𝑐_𝑥 is the constant map at 𝑥. Thus 𝐿^bord(𝑓)∶=[Fix(𝑓₁),𝑔,𝑔] determines the element in Ω_𝑘^𝑓𝑟(ℒ_𝑓𝑀;𝜖).

Applying Theorem 2, We obtain the following corollary.

Corollary 9 (converse of fiberwise Lefschetz fixed point theorem). Let 𝑓∶𝑀^𝑚+𝑘→𝑀^𝑚+𝑘 be a smooth bundle map over the closed manifold 𝐵^𝑘. Assume that 𝑚≥𝑘+3. Then, 𝑓 is fiber homotopic to a fixed point free map if and only if 𝐿^bord(𝑓)=0∈Ω_𝑘^𝑓𝑟(ℒ_𝑓𝑀;𝜖).

Corollary 10. Let 𝑓∶𝑀^𝑚+𝑘→𝑀^𝑚+𝑘 be a smooth bundle map over the closed manifold 𝐵^𝑘. Assume that 𝑚≥𝑘+3. If there is [𝑁,𝑁→−^𝑔ℒ_𝑓𝑀,𝜈_𝑁⊆𝑆^𝑚+𝑘𝑔−→𝜖]∈Ω_𝑘^𝑓𝑟(ℒ_𝑓𝑀;𝜖)𝑓 where 𝑁 is a finite subset of 𝑀 such that 𝐿^bord(𝑓)=[𝑁,𝑔,𝑔], then the map 𝑓 can be fiber-preserving homotoped to a map ℎ such 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.