Homotopic maps induce the same reduced-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 mathlib2Upstreamed into Mathlib.fully proved3Proved, and every prerequisite is fully proved too.

graph LR n0["Reduced singular homology"]:::fully_proved n1("Chain-homotopic maps induce the same homology map"):::mathlib n2["A topological homotopy gives a singular-chain homotopy"]:::mathlib n3("Homotopic maps induce the same reduced-homology map"):::fully_proved n4["A homotopy equivalence induces reduced-homology isomorphisms"]:::fully_proved n3 -.-> n4 n1 -.-> n3 n0 --> n4 n0 --> n3 n2 -.-> n3 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/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#topological-homotopy-chain-homotopy" "A topological homotopy gives a singular-chain homotopy — 1 item · 1 in mathlib" class n3 focus click n3 "../../../../../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 n4 "../../../../../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"