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Apr 17, 2019 at 14:37 vote accept Irene_233
Sep 5, 2018 at 9:13 comment added Russ Woodroofe I missed your confusion about which chain groups one is operating in. An important point is that C_k and C^k are isomorphic, and since C_k has a preferred basis (the k-faces), there is a preferred isomorphism between them. So the Laplacian maps are just "down-up + up-down" maps. A good place to read about (a slightly different version of) Laplacians is Simplicial matrix-tree theorems, by Duval, Klivans, and Martin.
Sep 3, 2018 at 13:10 history answered Russ Woodroofe CC BY-SA 4.0