Small singular chains include by a chain-homotopy equivalence: 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 proved6Proved, and every prerequisite is fully proved too.
graph LR
n0["Singular-chain subdivision is naturally homotopic to the identity"]:::fully_proved
n1("Iterated subdivision eventually makes every singular simplex small"):::fully_proved
n2["Small singular chains include by a chain-homotopy equivalence"]:::fully_proved
n3("The binary-cover excision chain map is a homotopy equivalence"):::fully_proved
n4("The binary-cover Mayer–Vietoris sequence"):::fully_proved
n5["Small augmented chains include by a chain-homotopy equivalence"]:::fully_proved
n3 --> n4
n1 --> n2
n0 --> n1
n0 -.-> n2
n2 -.-> n4
n2 --> n5
n2 -.-> 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#eventually-small-singular-simplices" "Iterated subdivision eventually makes every singular simplex small — 1 item · 1 fully proved"
class n2 focus
click n2 "../../../../../../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 n3 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#binary-cover-excision-chain-equivalence" "The binary-cover excision chain map is a homotopy equivalence — 1 item · 1 fully proved"
click n4 "../../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#binary-cover-mayer-vietoris-sequence" "The binary-cover Mayer–Vietoris sequence — 1 item · 1 fully proved"
click n5 "../../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#small-augmented-chain-inclusion-homotopy-equivalence" "Small augmented chains include by a chain-homotopy equivalence — 1 item · 1 fully proved"