Van Kampen's theorem 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 proved76Proved, and every prerequisite is fully proved too.
graph LR
n0["Attaching higher cells preserves the fundamental group"]:::fully_proved
n1["Attaching 2-cells adds the attaching relations"]:::fully_proved
n2("The attaching-sphere pieces generate the intersection group"):::fully_proved
n3["Binary van Kampen is a group pushout"]:::fully_proved
n4("An open-cover model for attached cells"):::fully_proved
n5["Cell-attachment support<br/><small>29 items · 29 fully proved</small>"]:::scope
n6["Cover factorizations of a loop"]:::fully_proved
n7["The group presentation associated to an open cover"]:::fully_proved
n8("Overlap relations vanish under the cover map"):::fully_proved
n9("Every group is a fundamental group"):::fully_proved
n10("Every group admits a generators-and-relations presentation"):::fully_proved
n11("Elementary factorization moves preserve the quotient class"):::fully_proved
n12["Finite skeleta above dimension two have the same fundamental group"]:::fully_proved
n13("Homotopic factorizations have the same quotient class"):::fully_proved
n14("A homotopy admits a threefold-incidence cover decomposition"):::fully_proved
n15["The pointed wedge of a family of spaces"]:::fully_proved
n16["Presentation-complex support<br/><small>13 items · 13 fully proved</small>"]:::scope
n17["The 2-skeleton determines the fundamental group"]:::fully_proved
n18("The kernel is generated by overlap relations"):::fully_proved
n19["Van Kampen's quotient isomorphism"]:::fully_proved
n20("The van Kampen cover map is surjective"):::fully_proved
n21["The fundamental group of a wedge of circles"]:::fully_proved
n22["The fundamental group of a wedge"]:::fully_proved
n23("The standard cover of a well-pointed wedge"):::fully_proved
n24["External chapter: Appendix<br/><small>2 items · 2 fully proved</small>"]:::boundary
n25["External chapter: The fundamental group<br/><small>4 items · 4 fully proved</small>"]:::boundary
n26["External chapter: Homology<br/><small>6 items · 6 fully proved</small>"]:::boundary
n24 -.-> n12
n24 -.-> n17
n24 -. 2 .-> n5
n25 -.-> n0
n25 -->|2| n1
n25 -.-> n20
n25 --> n21
n25 -->|3| n5
n25 -. 2 .-> n5
n0 -.-> n12
n1 -->|3| n16
n2 -.-> n0
n2 --> n1
n3 -. 2 .-> n5
n4 -.-> n0
n4 --> n1
n4 --> n2
n6 --> n11
n6 --> n13
n7 --> n6
n7 --> n8
n7 --> n11
n7 --> n13
n7 --> n18
n7 --> n20
n8 -.-> n18
n8 --> n19
n10 -.-> n9
n11 -.-> n13
n12 -.-> n17
n13 -.-> n18
n14 -.-> n13
n15 --> n26
n15 --> n22
n15 --> n23
n15 -->|3| n16
n18 -.-> n19
n19 -.-> n3
n19 -.-> n22
n20 -.-> n2
n20 -.-> n19
n21 -->|2| n16
n22 --> n21
n23 --> n26
n23 -.-> n26
n23 --> n22
n5 --> n26
n5 -. 4 .-> n26
n5 -. 3 .-> n0
n5 -->|5| n1
n5 -->|2| n4
n5 -. 7 .-> n4
n5 -. 2 .-> n12
n5 --> n21
n5 -->|11| n16
n16 -.-> n26
n16 -.-> n9
click n0 "../../../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 n1 "../../../roadmap/fundamental-group/van-kampen/index.html#attach-two-cells-fundamental-group" "Attaching 2-cells adds the attaching relations — 1 item · 1 fully proved"
click n2 "../../../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 n3 "../../../roadmap/fundamental-group/van-kampen/index.html#binary-van-kampen-pushout" "Binary van Kampen is a group pushout — 1 item · 1 fully proved"
click n4 "../../../roadmap/fundamental-group/van-kampen/index.html#cell-attachment-cover-model" "An open-cover model for attached cells — 1 item · 1 fully proved"
click n5 "van-kampen/cell-attachment-support.html" "Cell-attachment support — 29 items · 29 fully proved"
click n6 "../../../roadmap/fundamental-group/van-kampen/index.html#cover-factorization" "Cover factorizations of a loop — 1 item · 1 fully proved"
click n7 "../../../roadmap/fundamental-group/van-kampen/index.html#cover-group-presentation" "The group presentation associated to an open cover — 1 item · 1 fully proved"
click n8 "../../../roadmap/fundamental-group/van-kampen/index.html#cover-map-relations" "Overlap relations vanish under the cover map — 1 item · 1 fully proved"
click n9 "../../../roadmap/fundamental-group/van-kampen/index.html#every-group-fundamental-group" "Every group is a fundamental group — 1 item · 1 fully proved"
click n10 "../../../roadmap/fundamental-group/van-kampen/index.html#every-group-presentation" "Every group admits a generators-and-relations presentation — 1 item · 1 fully proved"
click n11 "../../../roadmap/fundamental-group/van-kampen/index.html#factorization-moves" "Elementary factorization moves preserve the quotient class — 1 item · 1 fully proved"
click n12 "../../../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"
click n13 "../../../roadmap/fundamental-group/van-kampen/index.html#homotopic-factorizations" "Homotopic factorizations have the same quotient class — 1 item · 1 fully proved"
click n14 "../../../roadmap/fundamental-group/van-kampen/index.html#homotopy-cover-decomposition" "A homotopy admits a threefold-incidence cover decomposition — 1 item · 1 fully proved"
click n15 "../../../roadmap/fundamental-group/van-kampen/index.html#pointed-wedge" "The pointed wedge of a family of spaces — 1 item · 1 fully proved"
click n16 "van-kampen/presentation-complex.html" "Presentation-complex support — 13 items · 13 fully proved"
click n17 "../../../roadmap/fundamental-group/van-kampen/index.html#two-skeleton-fundamental-group" "The 2-skeleton determines the fundamental group — 1 item · 1 fully proved"
click n18 "../../../roadmap/fundamental-group/van-kampen/index.html#van-kampen-kernel" "The kernel is generated by overlap relations — 1 item · 1 fully proved"
click n19 "../../../roadmap/fundamental-group/van-kampen/index.html#van-kampen-quotient" "Van Kampen's quotient isomorphism — 1 item · 1 fully proved"
click n20 "../../../roadmap/fundamental-group/van-kampen/index.html#van-kampen-surjective" "The van Kampen cover map is surjective — 1 item · 1 fully proved"
click n21 "../../../roadmap/fundamental-group/van-kampen/index.html#wedge-circles-free-group" "The fundamental group of a wedge of circles — 1 item · 1 fully proved"
click n22 "../../../roadmap/fundamental-group/van-kampen/index.html#wedge-fundamental-group" "The fundamental group of a wedge — 1 item · 1 fully proved"
click n23 "../../../roadmap/fundamental-group/van-kampen/index.html#well-pointed-wedge-cover" "The standard cover of a well-pointed wedge — 1 item · 1 fully proved"
click n24 "../../chapters/appendix.html" "Appendix — 2 items · 2 fully proved"
click n25 "../../chapters/fundamental-group.html" "The fundamental group — 4 items · 4 fully proved"
click n26 "../../chapters/homology.html" "Homology — 6 items · 6 fully proved"