Attaching higher cells preserves the fundamental group: 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 proved8Proved, and every prerequisite is fully proved too.

graph LR n0("Higher spheres are simply connected"):::fully_proved n1["The auxiliary cover basepoint transports to the original basepoint"]:::fully_proved n2("An open-cover model for attached cells"):::fully_proved n3["An attachment with no cells is an isomorphism"]:::fully_proved n4("The attaching-sphere pieces generate the intersection group"):::fully_proved n5["A binary cover with a contractible second piece"]:::fully_proved n6["Attaching higher cells preserves the fundamental group"]:::fully_proved n7["Finite skeleta above dimension two have the same fundamental group"]:::fully_proved n0 -.-> n6 n6 -.-> n7 n4 -.-> n6 n2 -.-> n6 n2 --> n4 n1 -.-> n6 n1 -.-> n2 n5 -.-> n6 n3 -.-> n6 click n0 "../../../../roadmap/fundamental-group/basic-constructions/index.html#sphere-simply-connected" "Higher spheres are simply connected — 1 item · 1 fully proved" click n1 "../../../../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 n2 "../../../../roadmap/fundamental-group/van-kampen/index.html#cell-attachment-cover-model" "An open-cover model for attached cells — 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" click n4 "../../../../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 n5 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#binary-cover-contractible-piece" "A binary cover with a contractible second piece — 1 item · 1 fully proved" class n6 focus click n6 "../../../../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 n7 "../../../../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"