A Bordism Viewpoint of Fiberwise Intersections

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 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 . 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 . 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 [4]). ？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)