I am studying the concepts of finite simplicial complexes (so please forgive for any mistakes :)) and, in particular, trying to grasp the concept of higher-order connectedness.
I have the following intuition from graph theory. We can define connected graphs in the following two equivalent ways (see Is this an equivalent definition for connectivity in graphs? )
- $G$ is connected iff for all pairs of vertices $u,v \in V$, there exists a path between $u$ and $v$.
- $G$ is connected iff for all partitions $\{A,B\}$ of $V(G)$ in two non-empty parts, there exists an edge with one endpoint in $A$ and the other in $B$.
My impression is that, using a topological point of view, we can view a graph $G$ as a $1$-dimensional simplicial complex, and "graph connectivity" is equivalent to the $0$-connectivity (or path connectivity) of this complex, i.e. for each continuous map $f: S^0 \rightarrow G$ can be extended to a continuous map $\overline{f}: B^1 \rightarrow G$. The parallel is clear with the first definition of graph connectivity.
Now, considering $1$-connectivity in a finite simplicial complex $X$ (i.e. each continuous map $f: S^1 \rightarrow X$ can be extended to a continuous map $\overline{f}: B^2 \rightarrow X$), I wonder if there is "more combinatorial" way of expressing that property in such a way that is more similar to the definitions for graph-connectivity I gave earlier.
For instance, I kind of visualize this property as (being vague here) saying that any "cycle" $C$ in $X$ we can find some "surface" in $X$ with $C$ as its border. But at the moment I don't imagine something which is analogous to the definition of connectivity in graphs in terms of its "cuts", that is, a partition of the vertices which "separate" the graph somehow.
So my question is, does such a "dual" definition of $1$-connectivity exist? If so, how it is formalised? Can it be understood in combinatorial terms like the cuts in graphs?
The usual "$1$-connectedness" analogue of your property 1 is that any cycle of edges in the 1-skeleton extends to a Van Kampen diagram.
But I don't know of an analogue of your property 2.