Homotopy type of an $n$-simplex with a $k$-face removed

Question

This must be an easy exercise but somehow I do not see an immediate answer. For natural numbers $k\leqslant n$, what is the homotopy type of the simplicial complex whose simplices are those (nonempty) subsets $\sigma\subset\{1,...,n\}$ with $\{1,...,k\}\nsubseteq\sigma$?

Answer 1

For $k=n$ it's the $(n-2)$-sphere, this could even be the definition of a sphere.

For $k<n$ it's contractible because the geometric realization is star-convex: the map $(x,t)\mapsto (1-t)x+(0,0,\dots,0,t)$ is a strong deformation retract to $\{(0,0,\dots,0,1)\}.$