A homotopy equivalence induces reduced-homology isomorphisms: 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 proved6Proved, and every prerequisite is fully proved too.
graph LR
n0["Reduced singular homology"]:::fully_proved
n1("Homotopic maps induce the same reduced-homology map"):::fully_proved
n2["A homotopy equivalence induces reduced-homology isomorphisms"]:::fully_proved
n3("Reduced homology of a contractible space vanishes"):::fully_proved
n4("The reduced neighborhood-retract Mayer–Vietoris sequence"):::fully_proved
n5("Local homology of positive-dimensional Euclidean space"):::fully_proved
n2 -.-> n5
n2 -.-> n3
n2 -.-> n4
n1 -.-> n2
n0 --> n3
n0 --> n2
n0 --> n1
click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#reduced-homology-homotopy-invariance" "Homotopic maps induce the same reduced-homology map — 1 item · 1 fully proved"
class n2 focus
click n2 "../../../../../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 n3 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#contractible-space-reduced-homology" "Reduced homology of a contractible space vanishes — 1 item · 1 fully proved"
click n4 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#reduced-neighborhood-mayer-vietoris" "The reduced neighborhood-retract Mayer–Vietoris sequence — 1 item · 1 fully proved"
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"