The identity simplex generates relative simplex 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 proved9Proved, and every prerequisite is fully proved too.

graph LR n0["The boundary and zero horn of a standard topological simplex"]:::fully_proved n1["A standard simplex strongly deformation retracts onto its zero horn"]:::fully_proved n2("Homology of a point"):::fully_proved n3("The zero face induces an isomorphism in relative homology"):::fully_proved n4("The long exact sequence of a triple"):::fully_proved n5("The triple connecting map sends a relative cycle to its boundary"):::fully_proved n6("Homotopic maps of pairs induce the same relative-homology map"):::fully_proved n7("The identity simplex generates relative simplex homology"):::fully_proved n8("The double-simplex difference generates reduced homology"):::fully_proved n7 -.-> n8 n0 --> n7 n0 --> n3 n0 --> n1 n3 -.-> n7 n1 -.-> n7 n5 -.-> n7 n2 -.-> n7 n6 -.-> n7 n4 -.-> n7 n4 -.-> n5 click n0 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-boundary-horn" "The boundary and zero horn of a standard topological simplex — 1 item · 1 fully proved" click n1 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-horn-retraction" "A standard simplex strongly deformation retracts onto its zero horn — 1 item · 1 fully proved" click n2 "../../../../../roadmap/homology/simplicial-and-singular/index.html#point-homology" "Homology of a point — 1 item · 1 fully proved" click n3 "../../../../../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 n4 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#triple-long-exact-sequence" "The long exact sequence of a triple — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#triple-connecting-cycle-formula" "The triple connecting map sends a relative cycle to its boundary — 1 item · 1 fully proved" click n6 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-homology-homotopy-invariance" "Homotopic maps of pairs induce the same relative-homology map — 1 item · 1 fully proved" class n7 focus click n7 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#relative-simplex-fundamental-class" "The identity simplex generates relative simplex homology — 1 item · 1 fully proved" click n8 "../../../../../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"