Singular-chain subdivision is naturally homotopic to the identity: 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 mathlib1Upstreamed into Mathlib.fully proved5Proved, and every prerequisite is fully proved too.
graph LR
n0["The singular chain complex"]:::mathlib
n1("Affine barycentric subdivision is compatible with faces"):::fully_proved
n2["Affine-chain subdivision is homotopic to the identity"]:::fully_proved
n3["Singular-chain subdivision is naturally homotopic to the identity"]:::fully_proved
n4("Iterated subdivision eventually makes every singular simplex small"):::fully_proved
n5["Small singular chains include by a chain-homotopy equivalence"]:::fully_proved
n1 --> n2
n1 --> n3
n2 -.-> n3
n4 --> n5
n3 --> n4
n3 -.-> n5
n0 --> n3
click n0 "../../../../../../roadmap/homology/simplicial-and-singular/index.html#singular-chain-complex" "The singular chain complex — 1 item · 1 in mathlib"
click n1 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#affine-barycentric-subdivision" "Affine barycentric subdivision is compatible with faces — 1 item · 1 fully proved"
click n2 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#affine-chain-subdivision-homotopy" "Affine-chain subdivision is homotopic to the identity — 1 item · 1 fully proved"
class n3 focus
click n3 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#singular-chain-subdivision-homotopy" "Singular-chain subdivision is naturally homotopic to the identity — 1 item · 1 fully proved"
click n4 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#eventually-small-singular-simplices" "Iterated subdivision eventually makes every singular simplex small — 1 item · 1 fully proved"
click n5 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#small-chain-inclusion-homotopy-equivalence" "Small singular chains include by a chain-homotopy equivalence — 1 item · 1 fully proved"