Chain-homotopic maps induce the same homology map: 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. in mathlib3Upstreamed into Mathlib.fully proved3Proved, and every prerequisite is fully proved too.
graph LR
n0["A chain map induces a map on homology"]:::mathlib
n1("Chain-homotopic maps induce the same homology map"):::mathlib
n2("Homotopic maps induce the same singular-homology map"):::mathlib
n3["Binary-cover excision induces homology isomorphisms"]:::fully_proved
n4("Homotopic maps of pairs induce the same relative-homology map"):::fully_proved
n5("Homotopic maps induce the same reduced-homology map"):::fully_proved
n1 -.-> n5
n1 -.-> n4
n1 -.-> n3
n1 -.-> n2
n0 --> n1
n0 --> n2
click n0 "../../../../roadmap/homology/simplicial-and-singular/index.html#chain-map-homology" "A chain map induces a map on homology — 1 item · 1 in mathlib"
class n1 focus
click n1 "../../../../roadmap/homology/simplicial-and-singular/index.html#chain-homotopy-invariance" "Chain-homotopic maps induce the same homology map — 1 item · 1 in mathlib"
click n2 "../../../../roadmap/homology/simplicial-and-singular/index.html#singular-homology-homotopy-invariance" "Homotopic maps induce the same singular-homology map — 1 item · 1 in mathlib"
click n3 "../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#binary-cover-excision-homology-isomorphism" "Binary-cover excision induces homology isomorphisms — 1 item · 1 fully proved"
click n4 "../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-homology-homotopy-invariance" "Homotopic maps of pairs induce the same relative-homology map — 1 item · 1 fully proved"
click n5 "../../../../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"