Local homology is unchanged on an open neighborhood: local context
Project map · Chapter map · Full theorem DAG · Open textbook statement
This local map shows one dependency hop in either direction. The highlighted item is the current focus. fully proved8Proved, and every prerequisite is fully proved too.
graph LR
n0["Relative singular chains and homology"]:::fully_proved
n1["Deleted-subset excision induces homology isomorphisms"]:::fully_proved
n2("Local homology is unchanged on an open neighborhood"):::fully_proved
n3("Deleted-pair homology is local on an open neighborhood"):::fully_proved
n4["Disjoint punctured neighborhoods form the deleted union pair"]:::fully_proved
n5["An isolated preimage gives a map of punctured pairs"]:::fully_proved
n6("Local homology of positive-dimensional Euclidean space"):::fully_proved
n7("Homeomorphic nonempty Euclidean open sets have equal dimension"):::fully_proved
n2 --> n3
n2 --> n4
n2 --> n5
n2 -.-> n7
n2 --> n6
n6 -.-> n7
n0 --> n3
n0 --> n2
n0 --> n1
n1 -.-> n2
click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#deleted-subset-excision-homology-isomorphism" "Deleted-subset excision induces homology isomorphisms — 1 item · 1 fully proved"
class n2 focus
click n2 "../../../../../roadmap/homology/computations-and-applications/local-homology-and-dimension/index.html#local-homology-open-neighborhood" "Local homology is unchanged on an open neighborhood — 1 item · 1 fully proved"
click n3 "../../../../../roadmap/homology/computations-and-applications/local-degree/index.html#deleted-pair-open-neighborhood" "Deleted-pair homology is local on an open neighborhood — 1 item · 1 fully proved"
click n4 "../../../../../roadmap/homology/computations-and-applications/local-degree/index.html#disjoint-punctured-pair-iso" "Disjoint punctured neighborhoods form the deleted union pair — 1 item · 1 fully proved"
click n5 "../../../../../roadmap/homology/computations-and-applications/local-degree/index.html#isolated-preimage-neighborhood" "An isolated preimage gives a map of punctured pairs — 1 item · 1 fully proved"
click n6 "../../../../../roadmap/homology/computations-and-applications/local-homology-and-dimension/index.html#punctured-euclidean-local-homology" "Local homology of positive-dimensional Euclidean space — 1 item · 1 fully proved"
click n7 "../../../../../roadmap/homology/computations-and-applications/local-homology-and-dimension/index.html#invariance-of-dimension-open-sets" "Homeomorphic nonempty Euclidean open sets have equal dimension — 1 item · 1 fully proved"