Reduced homology of a contractible space vanishes: 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 proved10Proved, and every prerequisite is fully proved too.

graph LR n0["Reduced singular homology"]:::fully_proved n1["Relative homology at a basepoint is reduced homology"]:::fully_proved n2["A homotopy equivalence induces reduced-homology isomorphisms"]:::fully_proved n3("Reduced homology of a contractible space vanishes"):::fully_proved n4["A contractible subspace identifies reduced and relative homology"]:::fully_proved n5("A nonsurjective sphere map has degree zero"):::fully_proved n6["Relative homology in a contractible ambient space"]:::fully_proved n7("The reduced homology of spheres satisfies the suspension recurrence"):::fully_proved n8("Reduced homology of a wedge is the direct sum of the summands"):::fully_proved n9("The boundary sphere is not a retract of the disk"):::fully_proved n3 -.-> n5 n3 -.-> n6 n3 -.-> n4 n3 -.-> n8 n3 -.-> n9 n3 -.-> n7 n2 -.-> n3 n1 --> n8 n1 -.-> n3 n0 --> n4 n0 --> n3 n0 --> n2 n0 --> n7 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/simplicial-and-singular/relative-homology/index.html#pointed-relative-reduced-homology" "Relative homology at a basepoint is reduced homology — 1 item · 1 fully proved" 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" class n3 focus 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/post-excision-applications/index.html#contractible-subspace-relative-reduced-homology" "A contractible subspace identifies reduced and relative homology — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#nonsurjective-degree-zero" "A nonsurjective sphere map has degree zero — 1 item · 1 fully proved" click n6 "../../../../../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 n7 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#sphere-reduced-homology-recurrence" "The reduced homology of spheres satisfies the suspension recurrence — 1 item · 1 fully proved" click n8 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#wedge-reduced-homology" "Reduced homology of a wedge is the direct sum of the summands — 1 item · 1 fully proved" click n9 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#disk-boundary-no-retraction" "The boundary sphere is not a retract of the disk — 1 item · 1 fully proved"