An open-cover model for attached cells: 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 proved13Proved, and every prerequisite is fully proved too.
graph LR
n0["Every abstract cell attachment has an indexed cone model"]:::fully_proved
n1["The strip-enlarged cell-attachment space"]:::fully_proved
n2["The strip enlargement retracts onto the attached space"]:::fully_proved
n3("Hatcher's binary cover of the strip enlargement"):::fully_proved
n4["The base-side auxiliary cover retracts onto the original space"]:::fully_proved
n5("The strip-connected upper cover is contractible"):::fully_proved
n6["The auxiliary cover basepoint transports to the original basepoint"]:::fully_proved
n7("The auxiliary overlap has an indexed open cover"):::fully_proved
n8["Each overlap piece has the homotopy type of its attaching sphere"]:::fully_proved
n9("An open-cover model for attached cells"):::fully_proved
n10("The attaching-sphere pieces generate the intersection group"):::fully_proved
n11["Attaching higher cells preserves the fundamental group"]:::fully_proved
n12["Attaching 2-cells adds the attaching relations"]:::fully_proved
n10 -.-> n11
n10 --> n12
n9 -.-> n11
n9 --> n12
n9 --> n10
n0 --> n9
n2 -.-> n9
n4 -.-> n9
n4 --> n6
n3 -.-> n9
n3 --> n4
n3 --> n6
n3 --> n7
n3 --> n5
n1 --> n9
n1 --> n2
n1 --> n3
n6 -.-> n11
n6 --> n12
n6 -.-> n9
n7 -.-> n9
n7 --> n8
n8 -.-> n9
n5 -.-> n9
click n0 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#abstract-cell-attachment-indexed-model" "Every abstract cell attachment has an indexed cone model — 1 item · 1 fully proved"
click n1 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-cell-attachment-space" "The strip-enlarged cell-attachment space — 1 item · 1 fully proved"
click n2 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-attachment-retract" "The strip enlargement retracts onto the attached space — 1 item · 1 fully proved"
click n3 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-cell-attachment-open-cover" "Hatcher's binary cover of the strip enlargement — 1 item · 1 fully proved"
click n4 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-base-cover-retract" "The base-side auxiliary cover retracts onto the original space — 1 item · 1 fully proved"
click n5 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-upper-cover-contractible" "The strip-connected upper cover is contractible — 1 item · 1 fully proved"
click n6 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-cover-basepoint-transport" "The auxiliary cover basepoint transports to the original basepoint — 1 item · 1 fully proved"
click n7 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-overlap-open-cover" "The auxiliary overlap has an indexed open cover — 1 item · 1 fully proved"
click n8 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-overlap-piece-sphere" "Each overlap piece has the homotopy type of its attaching sphere — 1 item · 1 fully proved"
class n9 focus
click n9 "../../../../roadmap/fundamental-group/van-kampen/index.html#cell-attachment-cover-model" "An open-cover model for attached cells — 1 item · 1 fully proved"
click n10 "../../../../roadmap/fundamental-group/van-kampen/index.html#attaching-spheres-generate-intersection" "The attaching-sphere pieces generate the intersection group — 1 item · 1 fully proved"
click n11 "../../../../roadmap/fundamental-group/van-kampen/index.html#attach-higher-cells-fundamental-group" "Attaching higher cells preserves the fundamental group — 1 item · 1 fully proved"
click n12 "../../../../roadmap/fundamental-group/van-kampen/index.html#attach-two-cells-fundamental-group" "Attaching 2-cells adds the attaching relations — 1 item · 1 fully proved"