Local homology of positive-dimensional Euclidean space: 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. What the colours mean

fully proved7Proved, and every prerequisite is fully proved too.

graph LR n0["A homotopy equivalence induces reduced-homology isomorphisms"]:::fully_proved n1("Local homology is unchanged on an open neighborhood"):::fully_proved n2["Relative homology in a contractible ambient space"]:::fully_proved n3["A sphere strongly deformation retracts from punctured Euclidean space"]:::fully_proved n4("Reduced homology of spheres"):::fully_proved n5("Local homology of positive-dimensional Euclidean space"):::fully_proved n6("Homeomorphic nonempty Euclidean open sets have equal dimension"):::fully_proved n2 -.-> n5 n1 -.-> n6 n1 --> n5 n5 -.-> n6 n0 -.-> n5 n3 -.-> n5 n4 -.-> n5 click n0 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#homotopy-equivalence-reduced-homology-iso" "A homotopy equivalence induces reduced-homology isomorphisms — 1 item · 1 fully proved" click n1 "../../../../../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 n2 "../../../../../roadmap/homology/computations-and-applications/local-homology-and-dimension/index.html#contractible-ambient-relative-homology" "Relative homology in a contractible ambient space — 1 item · 1 fully proved" click n3 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#punctured-space-radial-deformation-retract" "A sphere strongly deformation retracts from punctured Euclidean space — 1 item · 1 fully proved" click n4 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#sphere-reduced-homology" "Reduced homology of spheres — 1 item · 1 fully proved" class n5 focus click n5 "../../../../../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 n6 "../../../../../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"