Local degree and the sum over a finite fiber dependency map
This map shows the container's direct articles. Nested containers are collapsed and clickable; dependency edges are rolled up from their leaves. fully proved24Proved, and every prerequisite is fully proved too.ready to state2Prerequisites are stated, so this can be written down.planned5Described in the blueprint only.
graph LR
n0("Degree is the sum of the local degrees"):::planned
n1("Deleted-pair homology is local on an open neighborhood"):::fully_proved
n2["A disjoint open family is homeomorphic to its sigma type"]:::fully_proved
n3["Disjoint punctured neighborhoods form the deleted union pair"]:::fully_proved
n4("The finite-fiber pair map acts by the local degrees"):::fully_proved
n5["A finite sphere-map fiber has disjoint isolated neighborhoods"]:::fully_proved
n6["Finite-puncture homology is the coproduct of the local groups"]:::planned
n7("The finite-puncture class is the sum of the local classes"):::planned
n8("The disjoint local pairs excise to the finite-puncture pair"):::fully_proved
n9("Finite-puncture projections have the diagonal matrix"):::planned
n10("The finite-fiber pair map carries the global degree"):::fully_proved
n11("A local homeomorphism has local degree plus or minus one"):::can_state
n12["An isolated preimage gives a map of punctured pairs"]:::fully_proved
n13("Local degree is unchanged when the neighborhood shrinks"):::fully_proved
n14("Local degree is independent of its neighborhood"):::fully_proved
n15("Local degree on a chosen isolated neighborhood"):::fully_proved
n16["Relative chains commute with coproducts of arbitrary pairs"]:::can_state
n17["Relative homology commutes with coproducts of arbitrary pairs"]:::planned
n18["The coproduct of a family of topological pairs"]:::fully_proved
n19["The ordered sphere class orients local homology"]:::fully_proved
n20["External chapter: Homology<br/><small>11 items · 11 fully proved</small>"]:::boundary
n20 -->|2| n1
n20 -.-> n1
n20 --> n3
n20 --> n10
n20 --> n11
n20 --> n12
n20 --> n16
n20 -.-> n16
n20 --> n17
n20 --> n18
n20 -->|3| n19
n1 --> n8
n2 --> n3
n3 --> n8
n4 --> n0
n5 --> n8
n5 --> n10
n6 --> n9
n7 --> n0
n8 --> n4
n8 --> n6
n9 --> n7
n10 --> n0
n10 --> n7
n12 --> n5
n12 --> n15
n13 --> n14
n14 --> n4
n14 --> n11
n15 --> n13
n16 --> n17
n17 --> n6
n18 --> n3
n18 --> n16
n19 --> n7
n19 --> n10
n19 --> n15
click n0 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#degree-local-sum-formula" "Degree is the sum of the local degrees — 1 item · 1 planned"
click n1 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#deleted-pair-open-neighborhood" "Deleted-pair homology is local on an open neighborhood — 1 item · 1 fully proved"
click n2 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#disjoint-open-union-homeomorphism" "A disjoint open family is homeomorphic to its sigma type — 1 item · 1 fully proved"
click n3 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#disjoint-punctured-pair-iso" "Disjoint punctured neighborhoods form the deleted union pair — 1 item · 1 fully proved"
click n4 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#finite-fiber-local-map" "The finite-fiber pair map acts by the local degrees — 1 item · 1 fully proved"
click n5 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#finite-fiber-neighborhoods" "A finite sphere-map fiber has disjoint isolated neighborhoods — 1 item · 1 fully proved"
click n6 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#finite-puncture-homology-coproduct" "Finite-puncture homology is the coproduct of the local groups — 1 item · 1 planned"
click n7 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#finite-puncture-orientation-sum" "The finite-puncture class is the sum of the local classes — 1 item · 1 planned"
click n8 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#finite-puncture-pair-excision" "The disjoint local pairs excise to the finite-puncture pair — 1 item · 1 fully proved"
click n9 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#finite-puncture-projection-matrix" "Finite-puncture projections have the diagonal matrix — 1 item · 1 planned"
click n10 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#global-degree-fiber-square" "The finite-fiber pair map carries the global degree — 1 item · 1 fully proved"
click n11 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#homeomorphism-local-degree" "A local homeomorphism has local degree plus or minus one — 1 item · 1 ready to state"
click n12 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#isolated-preimage-neighborhood" "An isolated preimage gives a map of punctured pairs — 1 item · 1 fully proved"
click n13 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#local-degree-shrinking" "Local degree is unchanged when the neighborhood shrinks — 1 item · 1 fully proved"
click n14 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#local-degree-well-defined" "Local degree is independent of its neighborhood — 1 item · 1 fully proved"
click n15 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#neighborhood-local-degree" "Local degree on a chosen isolated neighborhood — 1 item · 1 fully proved"
click n16 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#relative-chains-sigma-pair" "Relative chains commute with coproducts of arbitrary pairs — 1 item · 1 ready to state"
click n17 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#relative-homology-sigma-pair" "Relative homology commutes with coproducts of arbitrary pairs — 1 item · 1 planned"
click n18 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#sigma-topological-pair" "The coproduct of a family of topological pairs — 1 item · 1 fully proved"
click n19 "../../../../roadmap/homology/computations-and-applications/local-degree/index.html#sphere-local-orientation" "The ordered sphere class orients local homology — 1 item · 1 fully proved"
click n20 "../../../chapters/homology.html" "Homology — 11 items · 11 fully proved"