Ganea conjecture

Ganea's conjecture is a now disproved claim in algebraic topology. It states that

$\mathrm{cat}(X\times S^n)=\mathrm{cat}(X)+1$

for all $n>0$, where $\mathrm{cat}(X)$ is the Lusternik–Schnirelmann category of a topological space X, and Sⁿ is the n-dimensional sphere.

The inequality

$\mathrm{cat}(X\times Y)\le \mathrm{cat}(X)+\mathrm{cat}(Y)$

holds for any pair of spaces, $X$ and $Y$. Furthermore, $\mathrm{cat}(S^n)=1$, for any sphere $S^n$, $n>0$. Thus, the conjecture amounts to $\mathrm{cat}(X\times S^n)\ge \mathrm{cat}(X)+1$.

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 disproved a related conjecture, which stated that

$\mathrm{cat}(M\setminus \{p\})=\mathrm{cat}(M)-1,$

for a closed manifold $M$ and $p$ a point in $M$.

A minimum dimensional counterexample to the conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010. It has dimension 7 and $\mathrm{cat}(X)=2$, and for sufficiently large $n$, $\mathrm{cat}(X\times S^n)$ is also 2.

This work raises the question: For which spaces X is the Ganea condition, $\mathrm{cat}(X\times S^n)=\mathrm{cat}(X)+1$, satisfied? It has been conjectured that these are precisely the spaces X for which $\mathrm{cat}(X)$ equals a related invariant, $\mathrm{Qcat}(X).$Template:By whom

Furthermore, cat(X * S^n) = cat(X ⨇ S^n ⨧ Im Y + X Re X + Y) = 1 Im(X, Y), 1 Re(X, Y).

References

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• Zachary Marshall Gehring-Young.