Let $X$ be a simplicial complex and let $A,B\subset X$ be subcomplexes such that $C=A\cap B$ is a non-empty simplicial complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$, $C_{\cdot}(C)$ be the corresponding chain complexes which coefficients in a field $\mathbb{F}$.
Question: Under what conditions does there exist a chain map $m:C_{\cdot}(A)\to C_{\cdot}(B)$ such that $m$ restricts to the identity on $C_{\cdot}(C)$? What if $A,B,C,$ and $X$ are flag complexes?
Some Basic Observations:
It seems like simple sufficient condition is that there exist a retraction $r:A\to C$. However, this does not appear to be necessary. For instance, if we take
- $X$ to be the simplicial complex given by the faces $(1,2,4),(2,3,4),(3,1,4),(1,2,5),(2,3,5),(3,1,5)$
- $A$ to be the subcomplex given by the faces $(1,2,4),(2,3,4),(3,1,4)$
- $B$ to be the subcomplex given by the faces $(1,2,5),(2,3,5),(3,1,5)$
Then $C$ is the simplicial comlex given by the edges $(1,2)$, $(2,3)$, and $(1,3)$. While $C$ is not a retraction of $A$, an $m$ satisfying the above conditions comes from exchanging the vertices $4$ and $5$.
In general one can write down linear systems arising from the relevant commuting diagrams and solve them. The solutions haven't been very illuminating. However, if one considers small random complexes, then more often than not I have been able to solve the linear system. In a very non-rigorous way this seems to suggest that maybe there are some weak sufficient conditions for such maps to exist.
