I'm searching to learn more about any results or/and proofs (new or recent) about LScategory of Hspaces and that of coHspaces. Any references are welcomed Thanks
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 coHspace. This follows from the characterization of LScategory in terms of factorization of the iterated diagonal: let $X$ be a connected CWcomplex 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 ("LusternikSchnirelmann Category" by Cornea, Lupton, Oprea and Tanré). You can also have a look at James'paper: "Category,in the sense of LusternikSchnirelmann" Topology. Vol. 17. pp.331343 1978. LScategory of Lie groups is difficult to compute for example: $Cat_{LS}(SU(n))=n1$ (Singhof). We also know that $Cat_{LS}(Sp(3))=5$ (FernándezSuárez; GómezTato; Strom; Tanré "The LusternikSchnirelmann 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: MacíasVirgós; PereiraSáez; "An upper bound for the LusternikSchnirelmann 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 LScategory of Hspaces, take the $n$torus $T^n$ then $Cat_{LS}(T^n)=n$, a lower bound is given by the cuplength which is $n$ and an upper bound by the dimension which is also $n$. Also using cuplength 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 LScat determines coHspaces, I am not aware of any nice statements about LScat of $H$spaces (you can certainly say something about the rational LScategory of simplyconnected Hspaces). 

