Componentwise isomorphisms imply a relative isomorphism: 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

fully proved7Proved, and every prerequisite is fully proved too.

graph LR n0["Relative singular chains and homology"]:::fully_proved n1("The long exact sequence of a pair"):::fully_proved n2("Componentwise isomorphisms imply a relative isomorphism"):::fully_proved n3["Enlarging to the chosen neighborhood preserves relative homology"]:::fully_proved n4["Enlarging the quotient point preserves relative homology"]:::fully_proved n5("Excision holds for a union of CW subcomplexes"):::fully_proved n6["The mapping-cone cover has the relative homology of the original pair"]:::fully_proved n2 -.-> n5 n2 -.-> n6 n2 -.-> n3 n2 -.-> n4 n1 -.-> n2 n0 --> n5 n0 --> n6 n0 --> n2 n0 --> n1 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/index.html#pair-long-exact-sequence" "The long exact sequence of a pair — 1 item · 1 fully proved" class n2 focus click n2 "../../../../../../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 n3 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-neighborhood-relative-homology" "Enlarging to the chosen neighborhood preserves relative homology — 1 item · 1 fully proved" click n4 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#point-quotient-neighborhood-relative-homology" "Enlarging the quotient point preserves relative homology — 1 item · 1 fully proved" 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" click n6 "../../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-cover-relative-homology" "The mapping-cone cover has the relative homology of the original pair — 1 item · 1 fully proved"