Surgery Applications to a Generalized Rudyak Conjecture

Abstract

Rudyak's conjecture states that cat$(M)\ge$ cat$(N)$ given a degree one map $f:M\to N$ between closed manifolds. We generalize this conjecture to sectional category, and follow the methodology of [5] to get the following result: Given a normal map of degree one $f:M\to N$ between smooth closed manifolds, fibrations $p^M:E^M\to M$ and $p^N:E^N\to N$, and lift $\overline{f}$ of $f$ with respect to $p^M$ and $p^N$, i.e., $f{p}^M=\overline{f}{p}^N$; then if $f$ has no surgery obstructions and $N$ satisfies the inequality $5\le \dim N\le 2r$ secat$(p^N)-3$ (where the fiber of $p^N$ is $(r-2)$-connected for some $r\ge 1$), then secat$(p^M)\ge$secat$(p^N)$. Finally, we apply this result to the case of higher topological complexity when $N$ is simply connected.

submitted