Higher-cell attachments preserve path components: 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 proved7Proved, and every prerequisite is fully proved too.
graph LR
n0("An indexed family of cones has a two-set open cover"):::fully_proved
n1["Every abstract cell attachment has an indexed cone model"]:::fully_proved
n2["The base-side auxiliary cover retracts onto the original space"]:::fully_proved
n3["An attachment with no cells is an isomorphism"]:::fully_proved
n4("Higher-cell attachments preserve path components"):::fully_proved
n5("The 2-skeleton of a path-connected CW complex is path-connected"):::fully_proved
n6["Finite skeleta above dimension two have the same fundamental group"]:::fully_proved
n1 --> n3
n1 -.-> n4
n2 -.-> n4
n3 -.-> n4
n4 -.-> n5
n4 -.-> n6
n0 --> n3
n0 -.-> n4
n5 -.-> n6
click n0 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#indexed-cone-open-cover" "An indexed family of cones has a two-set open cover — 1 item · 1 fully proved"
click n1 "../../../../../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 n2 "../../../../../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 n3 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#empty-cell-attachment-iso" "An attachment with no cells is an isomorphism — 1 item · 1 fully proved"
class n4 focus
click n4 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#higher-cell-attachment-path-components" "Higher-cell attachments preserve path components — 1 item · 1 fully proved"
click n5 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#path-connected-two-skeleton" "The 2-skeleton of a path-connected CW complex is path-connected — 1 item · 1 fully proved"
click n6 "../../../../../roadmap/fundamental-group/van-kampen/index.html#finite-skeleton-fundamental-group" "Finite skeleta above dimension two have the same fundamental group — 1 item · 1 fully proved"