The ordered difference of the two simplices is a cycle: 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["The boundary and zero horn of a standard topological simplex"]:::fully_proved n1["Reduced singular homology"]:::fully_proved n2["The ordered difference of the two simplices is a cycle"]:::fully_proved n3("The first simplex computes homology relative to the second"):::fully_proved n4("The double-simplex difference generates reduced homology"):::fully_proved n5("Swapping the double simplex negates its fundamental class"):::fully_proved n6["The double simplex is a sphere"]:::fully_proved n4 --> n5 n4 --> n6 n2 --> n5 n2 --> n4 n2 --> n3 n2 --> n6 n3 -.-> n4 n0 --> n2 n1 --> n2 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/simplicial-and-singular/relative-homology/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved" class n2 focus click n2 "../../../../../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 n3 "../../../../../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 n4 "../../../../../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" click n5 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#double-simplex-swap-class" "Swapping the double simplex negates its fundamental class — 1 item · 1 fully proved" click n6 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-sphere-homeomorphism" "The double simplex is a sphere — 1 item · 1 fully proved"