The first simplex computes homology relative to the second: 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 proved8Proved, and every prerequisite is fully proved too.
graph LR
n0["The ordered difference of the two simplices is a cycle"]:::fully_proved
n1["Standard-simplex boundary and horn pairs are good"]:::fully_proved
n2["The zero face identifies the two simplex point quotients"]:::fully_proved
n3("Good-pair relative homology is reduced quotient homology"):::fully_proved
n4["The good-pair quotient comparison is natural"]:::fully_proved
n5("The zero face induces an isomorphism in relative homology"):::fully_proved
n6("The first simplex computes homology relative to the second"):::fully_proved
n7("The double-simplex difference generates reduced homology"):::fully_proved
n0 --> n7
n0 --> n6
n6 -.-> n7
n1 -.-> n6
n1 -.-> n5
n2 -.-> n6
n2 -.-> n5
n5 -.-> n6
n3 -.-> n6
n3 -.-> n5
n3 --> n4
n4 -.-> n6
n4 -.-> n5
click n0 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-cycle" "The ordered difference of the two simplices is a cycle — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-boundary-good-pairs" "Standard-simplex boundary and horn pairs are good — 1 item · 1 fully proved"
click n2 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-face-point-quotient" "The zero face identifies the two simplex point quotients — 1 item · 1 fully proved"
click n3 "../../../../../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 n4 "../../../../../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 n5 "../../../../../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"
class n6 focus
click n6 "../../../../../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"
click n7 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-class" "The double-simplex difference generates reduced homology — 1 item · 1 fully proved"