Surgery Approach to Rudyak's Conjecture

TL;DR

This paper employs surgery theory to prove a new inequality relating the Lusternik-Schnirelmann categories of certain closed manifolds connected by a degree-one normal map, advancing understanding in topological manifold classification.

Contribution

It introduces a novel surgical approach to establish a category inequality for manifolds connected by degree-one maps under specific connectivity and dimension conditions.

Findings

Proves that under given conditions, at M \u2265 at N.

Establishes a new inequality involving at and manifold dimensions.

Utilizes surgery techniques to relate topological invariants.

Abstract

Using the surgery we prove the following: THEOREM. Let $f:M \to N$ be a normal map of degree one between closed manifolds with $N$ being $(r-1)$-connected, $r\ge 1$. If $N$ satisfies the inequality $\dim N \leq 2r \cat N - 3$, then for the Lusternik-Schnirelmann category $\cat M \geq \cat N$ .