I am working through the exercises in Fomenko/Fuchs "Homotopical Topology" and I find myself stumped by the following exercise regarding the Lusternik-Schnirelmann (LS) category of a topological space $X$ (denoted $catX$).
By definition the LS-category of a topological space $X$ is the minimum number of open sets $U_i$ which:
(1) Cover $X$
(2) Each $U_i$ is contractible in $X$
I am asked to show that for any nonempty topological space $X$, $cat \Sigma X \leq 2$.
My thoughts so far: If $X$ is contractible then it is homotopy equivalent to a point, and the suspension of a point is just a line which is clearly contractible. Since the LS-category is a homotopy invariant we have that $\Sigma X$ also has $cat \Sigma X =1$.
In the event that $X$ is not a contractible space, I'm a little unsure as to how to proceed. I would assume that the proof might intuitively involve the upper and lower halves of the suspension in the sense that if we think of the suspension as the union of two cones of $X$, then each cone is an open set which retracts onto its vertex. I am having some trouble making this rigorous and any hints would be greatly appreciated.
Your intuitive approach is correct. If $X$ is contractible, then $\Sigma X$ is also contractible and $cat\Sigma X = 1$. In the general case note that $\Sigma X$ is the quotient of $X \times [-1,1]$ where bottom and top $X \times \lbrace \pm1 \rbrace$ are identified to the suspension points $s_{\pm1} \in \Sigma X$. Let $p : X \times [-1,1] \to \Sigma X$ denote the quotient map. The sets $C+ = p(X \times (-1/2,1])$ and $C- = p(X \times [-1,1/2))$ are open in $\Sigma X$. Both $C_{\pm1}$ are contractible to $s_{\pm1}$; therefore $cat\Sigma X \le 2$.
A contraction of $C_+$ is explicitly given by $H([x,s],t) = [x,(1-t)s + t]$; $C_-$ is treated similarly.