Homotopic maps induce the same singular-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. What the colours mean

in mathlib5Upstreamed into Mathlib.fully proved1Proved, and every prerequisite is fully proved too.

graph LR n0["A chain map induces a map on homology"]:::mathlib n1["The singular chain complex"]:::mathlib n2("Chain-homotopic maps induce the same homology map"):::mathlib n3["A topological homotopy gives a singular-chain homotopy"]:::mathlib n4("Homotopic maps induce the same singular-homology map"):::mathlib n5["A homotopy equivalence induces homology isomorphisms"]:::fully_proved n2 -.-> n4 n0 --> n2 n0 -.-> n5 n0 --> n4 n1 --> n4 n1 --> n3 n4 -.-> n5 n3 -.-> n4 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" click n1 "../../../../roadmap/homology/simplicial-and-singular/index.html#singular-chain-complex" "The singular chain complex — 1 item · 1 in mathlib" click n2 "../../../../roadmap/homology/simplicial-and-singular/index.html#chain-homotopy-invariance" "Chain-homotopic maps induce the same homology map — 1 item · 1 in mathlib" click n3 "../../../../roadmap/homology/simplicial-and-singular/index.html#topological-homotopy-chain-homotopy" "A topological homotopy gives a singular-chain homotopy — 1 item · 1 in mathlib" class n4 focus click n4 "../../../../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 n5 "../../../../roadmap/homology/simplicial-and-singular/index.html#homotopy-equivalence-homology-iso" "A homotopy equivalence induces homology isomorphisms — 1 item · 1 fully proved"