Good-pair relative homology is reduced quotient homology: 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 proved11Proved, and every prerequisite is fully proved too.

graph LR n0["Good pairs and neighborhood deformation retracts"]:::fully_proved n1["Functorial point quotients of topological pairs"]:::fully_proved n2["Relative homology at a basepoint is reduced homology"]:::fully_proved n3["Enlarging to the chosen neighborhood preserves relative homology"]:::fully_proved n4["The complementary excision diagram"]:::fully_proved n5["Enlarging the quotient point preserves relative homology"]:::fully_proved n6["The neighborhood quotient map preserves relative homology"]:::fully_proved n7("Good-pair relative homology is reduced quotient homology"):::fully_proved n8["The good-pair quotient comparison is natural"]:::fully_proved n9("The zero face induces an isomorphism in relative homology"):::fully_proved n10("The first simplex computes homology relative to the second"):::fully_proved n9 -.-> n10 n0 --> n3 n0 --> n7 n0 --> n8 n4 -.-> n7 n4 --> n6 n3 --> n4 n3 -.-> n7 n7 -.-> n10 n7 -.-> n9 n7 --> n8 n8 -.-> n10 n8 -.-> n9 n1 --> n7 n1 --> n8 n6 -.-> n7 n5 -.-> n7 n2 --> n7 click n0 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-data" "Good pairs and neighborhood deformation retracts — 1 item · 1 fully proved" click n1 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#point-quotient-functor" "Functorial point quotients of topological pairs — 1 item · 1 fully proved" click n2 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pointed-relative-reduced-homology" "Relative homology at a basepoint is reduced homology — 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#good-pair-excision-diagram" "The complementary excision diagram — 1 item · 1 fully proved" click n5 "../../../../../../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 n6 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#point-quotient-neighborhood-excision" "The neighborhood quotient map preserves relative homology — 1 item · 1 fully proved" class n7 focus click n7 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-point-quotient-relative-homology" "Good-pair relative homology is reduced quotient homology — 1 item · 1 fully proved" click n8 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-quotient-comparison-naturality" "The good-pair quotient comparison is natural — 1 item · 1 fully proved" click n9 "../../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-face-relative-homology" "The zero face induces an isomorphism in relative homology — 1 item · 1 fully proved" click n10 "../../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-relative-comparison" "The first simplex computes homology relative to the second — 1 item · 1 fully proved"