Iterated subdivision eventually makes every singular simplex small: 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 proved5Proved, and every prerequisite is fully proved too.
graph LR
n0["Singular-chain subdivision is naturally homotopic to the identity"]:::fully_proved
n1("Barycentric subdivision contracts simplex diameter"):::fully_proved
n2("Iterated subdivision eventually makes every singular simplex small"):::fully_proved
n3["Small singular chains include by a chain-homotopy equivalence"]:::fully_proved
n4["The binary-cover Mayer–Vietoris chain complex"]:::fully_proved
n1 -.-> n2
n2 --> n4
n2 --> n3
n0 --> n2
n0 -.-> n3
click n0 "../../../../../../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 n1 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#barycentric-subdivision-diameter" "Barycentric subdivision contracts simplex diameter — 1 item · 1 fully proved"
class n2 focus
click n2 "../../../../../../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 n3 "../../../../../../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"
click n4 "../../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#binary-cover-chain-complex" "The binary-cover Mayer–Vietoris chain complex — 1 item · 1 fully proved"