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2
votes
Combinatorial distance between simplicial complexes
This is a homotopy-theoretical version of your idea.
Metric Structures for CW Complexes
By Nicholas Scoville
Topology Proceedings, Volume 44 (2014) 117-131
3
votes
Removing simplices from simplicial complexes without decreasing connectedness
This is just another point of view on the already-accepted answer.
Especially when working homotopy-theoretically, it's hard to think about `removing a simplex', but we can think instead about addi …