Concept

Ganea conjecture

Summary
Ganea's conjecture is a now disproved claim in algebraic topology. It states that : \operatorname{cat}(X \times S^n)=\operatorname{cat}(X) +1 for all n>0, where \operatorname{cat}(X) is the of a topological space X, and Sn is the n-dimensional sphere. The inequality : \operatorname{cat}(X \times Y) \le \operatorname{cat}(X) +\operatorname{cat}(Y) holds for any pair of spaces, X and Y. Furthermore, \operatorname{cat}(S^n)=1, for any sphere S^n, n>0. Thus, the conjecture amounts to \operatorname{cat}(X \times S^n)\ge\operatorname{cat}(X) +1. The conjecture was formulated by Tudor Ganea in 1971. Many particular cases of this conjecture were proved, and Norio Iwase gave a counterexample to the general case in 1998. In a follow-up paper from 2002, Iwase gave an even stronger counterexample, with X a closed smooth manifold. This counterexample also
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