1 Answers
Ganea's conjecture is a claim in algebraic topology, now disproved. It states that
for all n > 0 {\displaystyle n>0} , where cat {\displaystyle \operatorname {cat} } is the Lusternik–Schnirelmann category of a topological space X, and S is the n-dimensional sphere.
The inequality
holds for any pair of spaces, X {\displaystyle X} and Y {\displaystyle Y}. Furthermore, cat = 1 {\displaystyle \operatorname {cat} =1} , for any sphere S n {\displaystyle S^{n}} , n > 0 {\displaystyle n>0}. Thus, the conjecture amounts to cat ≥ cat + 1 {\displaystyle \operatorname {cat} \geq \operatorname {cat} +1}.