Excision holds for a union of CW subcomplexes: 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["Relative singular chains and homology"]:::fully_proved
n1("Componentwise isomorphisms imply a relative isomorphism"):::fully_proved
n2["A homotopy equivalence induces homology isomorphisms"]:::fully_proved
n3["Binary-cover excision induces homology isomorphisms"]:::fully_proved
n4("Subcomplex decompositions admit compatible neighborhoods"):::fully_proved
n5("Excision holds for a union of CW subcomplexes"):::fully_proved
n4 -.-> n5
n2 -.-> n5
n1 -.-> n5
n0 --> n5
n0 --> n1
n0 --> n3
n3 -.-> n5
click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#relative-homology-map-isomorphism-criterion" "Componentwise isomorphisms imply a relative isomorphism — 1 item · 1 fully proved"
click n2 "../../../../../roadmap/homology/simplicial-and-singular/index.html#homotopy-equivalence-homology-iso" "A homotopy equivalence induces homology isomorphisms — 1 item · 1 fully proved"
click n3 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#binary-cover-excision-homology-isomorphism" "Binary-cover excision induces homology isomorphisms — 1 item · 1 fully proved"
click n4 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/cw-subcomplex-neighborhood/index.html#subcomplex-neighborhood-cover" "Subcomplex decompositions admit compatible neighborhoods — 1 item · 1 fully proved"
class n5 focus
click n5 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#cw-subcomplex-excision" "Excision holds for a union of CW subcomplexes — 1 item · 1 fully proved"