Surgery Applications to a Generalized Rudyak Conjecture

TL;DR

This paper extends Rudyak's conjecture from Lusternik-Schnirelmann category to sectional category, establishing inequalities under certain conditions involving degree-one maps, fibrations, and surgery obstructions, with applications to higher topological complexity.

Contribution

It generalizes Rudyak's conjecture to sectional category and provides new inequalities for fibrations under specific topological conditions.

Findings

Established secat$(p^M) \,\geq\, $secat$(p^N)$ under given conditions.

Extended the conjecture to higher topological complexity for simply connected manifolds.

Applied the results to cases with no surgery obstructions and specific dimensional bounds.

Abstract

Rudyak's conjecture states that cat$(M) \geq$ 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., $fp^M = \overline{f} p^N$; then if $f$ has no surgery obstructions and $N$ satisfies the inequality $5 \leq \dim N \leq 2r$ secat$(p^N) - 3$ (where the fiber of $p^N$ is $(r-2)$-connected for some $r \geq 1$), then secat$(p^M) \geq$secat$(p^N)$. Finally, we apply this result to the case of higher topological complexity when $N$ is simply connected.