A Bordism Viewpoint of Fiberwise Intersections

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 means the smooth manifold X of dimension a and I means the unit interval [0, 1]. If Y is a submanifold of X, f : X → Z, and η is a bundle over Z, then ]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. DefineΩ ∗ (X; ξ) to be the bordism groups of manifolds mapping toX, 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 Lbord(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 , and E M are smooth fiber