LS-cat of Hspaces and that of co-Hspaces

Question

I'm searching to learn more about any results or/and proofs (new or recent) about LS-category of H-spaces and that of co-Hspaces. Any references are welcomed Thanks

Answer 1

Let me use the normalized LS i.e.: $Cat_{LS}(S^n)=1$. By definition $Cat_{LS}(X)=0$ is equivalent to $X$ contractible. And $Cat_{LS}(X)=1$ corresponds to the case in which $X$ is a co-Hspace. This follows from the characterization of LS-category in terms of factorization of the iterated diagonal:

let $X$ be a connected CW-complex then $Cat_{LS}(X)\leq n$ if and only if the iterated diagonal map $X\rightarrow X^{n+1}$ factors through the fat wedge $T^{n+1}(X)$.

I think you will find a proof of this result in many places ("Lusternik-Schnirelmann Category" by Cornea, Lupton, Oprea and Tanré). You can also have a look at James'paper: "Category,in the sense of Lusternik-Schnirelmann" Topology. Vol. 17. pp.331-343 1978.

LS-category of Lie groups is difficult to compute for example: $Cat_{LS}(SU(n))=n-1$ (Singhof). We also know that $Cat_{LS}(Sp(3))=5$ (Fernández-Suárez; Gómez-Tato; Strom; Tanré "The Lusternik-Schnirelmann category of $\rm Sp(3)$" Proc. Amer. Math. Soc. 132 (2004), no. 2, 587–595).

To my knowledge we do not know the category of $Sp(n)$ when $n>3$, but we know some bounds: Macías-Virgós; Pereira-Sáez; "An upper bound for the Lusternik-Schnirelmann category of the symplectic group." Math. Proc. Cambridge Philos. Soc. 155 (2013), no. 2, 271–276. The LS category of the Lie group ${\rm Spin} (n)$ is known for $3\leq n \leq 9$.

Edit: Let me say more about LS-category of H-spaces, take the $n$-torus $T^n$ then $Cat_{LS}(T^n)=n$, a lower bound is given by the cup-length which is $n$ and an upper bound by the dimension which is also $n$. Also using cup-length you can prove that $Cat_{LS}(K(\mathbb{Z},2))$ is not finite, but on the other hand we know that $K(\mathbb{Q},1)=2$!!

If LS-cat determines co-Hspaces, I am not aware of any nice statements about LS-cat of $H$-spaces (you can certainly say something about the rational LS-category of simply-connected H-spaces).