Removing simplices from simplicial complexes without decreasing connectedness

Question

Let $X$ be a non-contractible, $(d-1)$-connected, $d$-dimensional simplicial complex. By the theorems of Hurewicz and Whitehead, $X$ is homotopy equivalent to a wedge of $d$-spheres. Does there exist a $d$-simplex that can be removed from $X$ without decreasing the connectedness?

Answer 1

For a counterexample, we glue two-dimensional Moore spaces for the groups $\mathbb Z/2$ and $\mathbb Z/3$ along a common $S^1$. As a CW complex, $X$ can be realised by gluing two 2-disks into a closed loop $\gamma$, such the boundaries of the disks wind around $\gamma$ two and three times, respectively.

To get a simplicial complex, start with a triangle, then glue in a simplicial 6-gon and a simplicial 9-gon as above (both need a sufficiently fine simplicial structure for this to work). Removing a 2-simplex produces a space that is homotopy equivalent the CW complex above with one of the two disks removed, that is, to one of the two Moore spaces.

The fundamental group has a generator $[\gamma]$ and two relations $2[\gamma]=0$ and $3[\gamma]=0$, so $X$ is simply connected. If you take away one of the two disks, you loose one relation, so you loose simple connectivity.

The associated cellular chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.

Answer 2

This is just another point of view on the already-accepted answer.

Especially when working homotopy-theoretically, it's hard to think about `removing a simplex', but we can think instead about adding a simplex (or cell). This leads to the reformulation: if $S^{n-1} \to X \to \bigvee S^n$ is a cofiber sequence (so $X$ is the result of subtracting the cell from the wedge), then what can you say about the connectivity of $X$?

For $n$ sufficiently large (probably $\geq 3$ or so) this is a stable question, so we can back up the cofiber sequence to get $\bigvee S^{n-1} \to S^{n-1} \to X\to \bigvee S^n$, and we simply need the map $\bigvee S^{n-1} \to S^{n-1}$ to be surjective on integer homology. This includes, in particular, the accepted example.