Homotopical intersection theory I

Abstract

We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [17], but our proofs are fundamentally different. Errata Minor errors were corrected on page 967 (18 February 2008).

Relative fixed point theory

“…In the next section we prove relative generalizations of the results in [14]. In this section we apply those results to relative fixed point invariants.…”

“…The approach of [14] is based on invariants that detect sections of fibrations. In the next section we prove relative generalizations of the results in [14].…”

“…Then, using functoriality, we show these invariants coincide. Finally, we generalize Klein and Williams' proof of the converse to the Lefschetz fixed point theorem in [14] to complete the proof of the converse to the relative Lefschetz fixed point theorem.…”

A manifold calculus approach to link maps and the linking number

“…Our work also intersects the very interesting recent work of Klein and Williams [13]. Section 9 of [13] specifically talks about linking and the linking number, and we would like to emphasize that our work on the linking number was done independently. Our work on the linking number also relates to the recent work of Chernov and Rudyak [1].…”

“…Consider the sequence of spaces F l → F t → ΩC 3 (P 1 , P 2 ; N). Theorem B of [13] shows that the latter map is (2n − p − q − 3)-connected, and Conjecture 34 implies the composed map has the desired connectivity. ✷…”