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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 430105, 6 pages
A Bordism Viewpoint of Fiberwise Intersections
50 Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
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.
In topological fixed point theory, there is a classical questions that people may ask; given a smooth self-map of a smooth compact manifold , when is 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 be a smooth self-map of a compact smooth manifold .
If has no fixed point, then the Lefschetz number , where .
For this work, we focus on the similar arguments as above for the family of smooth maps over a compact base space . The proof of the main theorem depends heavily on the intersection problem as follows.
From now on, the notations means the smooth manifold of dimension and means the unit interval . If is a submanifold of , , and is a bundle over , then means the normal bundle of in and is a pull-back bundle of along the map .
Later we define “framed bordism with coefficient in a bundle" as follows. Let be a smooth manifold with a bundle over it. Define to be the bordism groups of manifolds mapping to , together with a stable isomorphism of the normal bundle with the pullback of . This framed bordism group will be a home for our invariants (described in the last section) which detects more fixed point information than the regular Lefschetz number.
(I) Suppose that , , and are smooth fiber bundles over a compact manifold . Let be a bundle map, and let be a subbundle of with the inclusion bundle map .
We have a commutative diagram (1) where , , and are the fibers of , , and , respectively.
We may assume that in (see ).
The homotopy pullback is
We have a diagram which commutes up to homotopy (3) where and are the trivial projections; that is, we have a homotopy defined by , .
We also have a map defined by where constant path in at .
Transversality yields a bundle map (4)
Choose an embedding , for sufficiently large , where is a sphere of dimension . Then, we have . So .
The commutative diagram (5) yields a bundle map (6)
Thus, determines an element .
(II) Suppose that is another representative of , where . This means we have a normal bordism , between and ; that is (i), (ii), (iii), (iv) and such that
Theorem 2 (main theorem). Assume . Then, there exists a smooth fiber-preserving map over such that , and .
Note that if we let , then is fiber-preserving homotopic to and .
In 1974, Hatcher and Quinn  showed an interesting result as follows.
Theorem 3 (Hatcher-Quinn). Given smooth bundle maps and over a compact base space which are immersions in each fiber, assume and . Then there is a fiber-preserving map over homotopic to such that .
Their theorem required a map 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 ).
2. Proof of the Main Theorem
Theorem 4 (Whitney embedding theorem). Let and be smooth manifolds, and let be a smooth map. If , then is homotopic to an embedding .
Lemma 5. Let be a map between two smooth manifolds. Let be a closed submanifold of . Assume that is an embedding.
If , then is homotopic to an embedding relative to .
Proof. Let be a tubular neighborhood of in .
Step 1. Extend the embedding to an embedding .
Let be the normal bundle of in and denote the disc bundle of . Then the tubular neighborhood theorem implies .
Claim. For any given embedding and vector bundle over , then where is the normal bundle of in via .
Proof of Claim. We have a diagram (10) where is the exponential map.
Note that tubular neighborhood of in via . Then is a desired embedding.
Assume there exists an embedding so that the following diagram commutes (11)
We are in the situation where we have a commutative diagram (12)
Let . Then , where denotes the stable isomorphism between 2 vector bundles.
If , then , so there exists a bundle monomorphism (13)
Apply the Claim when and . Then we have an extension embedding of from .
Step 2. We have a map and .
The condition and Theorem 4 imply that is homotopic to an embedding .
Then is homotopic to relative to .
2.1. Proof of the Main Theorem
We divide the proof into steps.
Step 1. Goal: fiber-preserving homotoped the map to an embedding over .
By assumption, we have and we also have maps where , so , .
Recall that , is just the inclusion of and into .
Apply the condition to Lemma 5, there exists an embedding (rel ). That is, we have a commutative diagram (17)
We have a map . By concatenating the homotopy and together, we have a homotopy such that and . Hence, is homotopic to .
Next, we want to modify the homotopy such that it is fiber preserving with respect to .
Note that we have a commutative diagram (18)
We can apply the homotopy lifting property for to get a homotopy of to through such that the following diagram commute: (19)
Let . Then is a fiber-preserving map over through the lifting .
Step 2. Goal: construct a bundle isomorphism where is the trivial bundle.
Since , it is enough to give a stable equivalence between such bundles.
Now, we have
We also have a commutative diagram (22) Thus, the bundle map yields the following stable isomorphism Putting (21), (23), and (24) together, we get Consequently, we have This implies that we did construct a bundle map (27) which gives us the extension of map to the tubular neighborhood of in . More precisely, where denotes the disc bundle. Note that and .
Since , we can find a subbundle of such that . For simplicity, let .
Step 3. Goal: construct the fiber-preserving smooth map over .
Recall that we have
Then there exists a neighborhood of in such that and .
According to (19), we have a commutative diagram (30) where .
Let , .
According to (27), there exists a bundle over such that for .
Let , . Then is a homotopy equivalence for and also .
Since is a cofibration and a homotopy equivalence, there exist an extension such that the following diagram commutes (31)
Next we want to construct an embedding such that the following hold:(i), (ii),(iii).
We start by letting be a smooth map such that , and , and we also have an inclusion .
Let . Then is such a required map.
By the construction, we have
Let be the composition of the maps
Define a map .
Using the fact that , then is a cofibration and homotopy equivalence. Hence there exists an extension .
Note that for such that , the map forces that has to be in , so the definition of implies . Thus
We define a map by
Then is well-defined map over by the construction.
It’s not hard to see that is diffeomorphic to . Define the map to be the composition of maps where is the projection to the first factor.
Thus, we get a map over so that . By construction, and as required.
Corollary 6. Assume . Then the map can be fiber-preserving homotoped to a map whose image does not intersect if and only if .
3. Application to Fixed Point Theory
Let be a smooth fiber bundle with compact fibers and . Assume that is a closed manifold. Let be a smooth map over . That is, .
The fixed point set of is
We have a homotopy pull-back diagram (38) where (i), (ii) and are the evaluation map at and respectively, (iii) the diagonal map, defined by , (iv) the twisted diagonal map, defined by , (v) the fiber bundle over with fiber over , given by where is the fiber of over .
Proposition 7. There exists a homotopy from to such that .
The proof relies on the work of Koźniewski , relating to -manifolds. Let be a smooth manifold. A -manifold is a manifold together with a locally trivial submersion . A -map is a smooth fiber-preserving map.
Lemma 8. Let and be -manifolds, and let be a -submanifold of . Let be a -map. Then, there is a fiber-preserving smooth -homotopy such that and .
Proof. See  for the proof.
We have a transversal (pull-back) square (39) where is the inclusion. Transversality yields that .
Choose an embedding for sufficiently large . Then we have We denote this bundle isomorphism by . We also have a map defined by , where is the constant map at . Thus determines the element in .
Applying Theorem 2, We obtain the following corollary.
Corollary 9 (converse of fiberwise Lefschetz fixed point theorem). Let be a smooth bundle map over the closed manifold . Assume that . Then, is fiber homotopic to a fixed point free map if and only if .
Corollary 10. Let be a smooth bundle map over the closed manifold . Assume that . If there is where is a finite subset of such that , then the map can be fiber-preserving homotoped to a map such that .
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.
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