Suppose $X$ be a $k$-connected CW-complex. I want to know Is it possible to find the value of $k$ from the CW-structure? In other words, Suppose $X$ has $n_i$, $i$-cell. What is the maximum value of $k$ such that $X$ be a $k$-connected topological space?
2026-03-28 13:20:45.1774704045
CW Structure and connectedness.
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If $X$ has exactly one $0$-cell and no other cells until dimension $n$, then $X$ is $(n-1)$-connected. This is a consequence of the cellular approximation theorem.
Unfortunately this is the most you can say about $X$ without more information. For instance it is possible that $X$ may be more highly connected. As an example consider the contractible space $D^{n+1}=S^n\cup e^{n+1}$. To see now why it is impossible to read off the connectivity of an arbitrary CW complex from just its list of cells compare $D^{n+1}$ with $S^n\vee S^{n+1}$.