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?



Removing simplices from simplicial complexes without decreasing connectedness
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.


