Reduced singular homology: 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 proved15Proved, and every prerequisite is fully proved too.
graph LR
n0["The augmented singular chain complex"]:::fully_proved
n1["Reduced singular homology"]:::fully_proved
n2["The ordered difference of the two simplices is a cycle"]:::fully_proved
n3["Reduced and ordinary homology agree in positive degrees"]:::fully_proved
n4["Zeroth homology splits into reduced homology and the coefficient object"]:::fully_proved
n5["Relative homology at a basepoint is reduced homology"]:::fully_proved
n6("Homotopic maps induce the same reduced-homology map"):::fully_proved
n7["A homotopy equivalence induces reduced-homology isomorphisms"]:::fully_proved
n8("Reduced homology of a contractible space vanishes"):::fully_proved
n9["A contractible subspace identifies reduced and relative homology"]:::fully_proved
n10("The reduced binary-cover Mayer–Vietoris sequence"):::fully_proved
n11("The reduced homology of spheres satisfies the suspension recurrence"):::fully_proved
n12("Reduced homology of the zero-sphere"):::fully_proved
n13("Reduced homology of spheres"):::fully_proved
n14["Relative homology is reduced homology of the mapping cone"]:::fully_proved
n9 --> n14
n8 -.-> n9
n8 -.-> n11
n7 -.-> n8
n6 -.-> n7
n11 -.-> n13
n12 -.-> n13
n0 --> n1
n5 -.-> n8
n4 -.-> n12
n3 -.-> n12
n1 --> n10
n1 --> n9
n1 --> n2
n1 --> n14
n1 --> n8
n1 --> n7
n1 --> n6
n1 --> n13
n1 --> n11
n1 --> n12
n1 --> n5
n1 --> n4
n1 --> n3
click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#augmented-singular-chain-complex" "The augmented singular chain complex — 1 item · 1 fully proved"
class n1 focus
click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved"
click n2 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-cycle" "The ordered difference of the two simplices is a cycle — 1 item · 1 fully proved"
click n3 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-positive-degree-comparison" "Reduced and ordinary homology agree in positive degrees — 1 item · 1 fully proved"
click n4 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pointed-zeroth-homology-splitting" "Zeroth homology splits into reduced homology and the coefficient object — 1 item · 1 fully proved"
click n5 "../../../../../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 n6 "../../../../../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"
click n7 "../../../../../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 n8 "../../../../../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 n9 "../../../../../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 n10 "../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#reduced-binary-cover-mayer-vietoris-sequence" "The reduced binary-cover Mayer–Vietoris sequence — 1 item · 1 fully proved"
click n11 "../../../../../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 n12 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#zero-sphere-reduced-homology" "Reduced homology of the zero-sphere — 1 item · 1 fully proved"
click n13 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#sphere-reduced-homology" "Reduced homology of spheres — 1 item · 1 fully proved"
click n14 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-relative-reduced-homology" "Relative homology is reduced homology of the mapping cone — 1 item · 1 fully proved"