# Definition of k-Connectedness for Simplicial Complexes

Parroting the link below (I imagine this is a standard definition), a topological space $$T$$ is $$k$$-connected ($$k ≥ 0$$) if, for every $$0 ≤ r ≤ k$$, every continuous map $$f : S^r → T$$ extends to a continuous map $$f : B^{r+1} → T$$. ($$B^r$$ is the $$r$$-dimensional unit ball.) A simplicial complex is $$k$$-connected if its geometric realization is.

http://web.cs.elte.hu/~lovasz/kurzusok/topol13.pdf#page=2

Parroting the link below, two simplices $$\sigma$$ and $$\tau$$ are $$k$$-connected if there is a sequence of simplices $$\sigma$$; $$\sigma_1$$; $$\sigma_2$$; $$\dots$$; $$\sigma_n$$; $$\tau$$; such that any two consecutive simplices have at least $$k+1$$ vertices in common. A simplicial complex is $$k$$-connected if any two simplices of dimension greater than or equal to $$k$$ are $$k$$-connected.

https://math.la.asu.edu/~helene/papers/atheory_final.pdf#page=2

Are the above two definitions equivalent? If not, what is a more combinatorial way of describing what the first definition means?

They appear not to be equivalent. Take two tetrahedra (full complexes on four vertices) and join at a point. This gives us a simplicial complex on $$7$$ vertices whose geometric realisation is contractible.
By your first definition, this simplicial complex is $$2$$-connected. By your second, it appears not to be. It's not even $$1$$-connected.
Two simplices $$\sigma$$ and $$\tau$$ are $$k$$-connected if there is a sequence of simplices $$\sigma,...,\tau$$ such that any two consecutive simplices are joined by a simplex.