Fundamental groupoid

Seifert-Van Kampen-Brown theorem

Let be a topological space with open cover . Then the following is a fibre coproduct in on the left and in on the right.12 #m/thm/homotopy

https://q.uiver.app/#q=WzAsOCxbMCwwLCJVXFxjYXAgViJdLFswLDIsIlUiXSxbMiwwLCJWIl0sWzIsMiwiWCJdLFs0LDAsIlxccGlfMShVIFxcY2FwIFYpIl0sWzQsMiwiXFxwaV8xKFUpIl0sWzYsMiwiXFxwaV8xKFgpIl0sWzYsMCwiXFxwaV8xKFYpIl0sWzAsMiwiaV8yIl0sWzAsMSwiaV8xIiwyXSxbMSwzLCJqXzEiXSxbMiwzLCJqXzIiXSxbNCw3LCJcXHBpXzEgaV8yIl0sWzQsNSwiXFxwaV8xaV8xIiwyXSxbNSw2LCJcXHBpXzFqXzEiLDJdLFs3LDYsIlxccGlfMmpfMiJdLFs2LDQsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFszLDAsIiIsMCx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==

where denote natural inclusions; i.e. the Fundamental groupoid of is a fibre coproduct of the fundamental groupoids of the open covering spaces and .

Proof

For the left diagram see Fibre products and coproducts in Top. Now suppose fits into the following diagram.

https://q.uiver.app/#q=WzAsNSxbMCwwLCJcXHBpXzEoVSBcXGNhcCBWKSJdLFswLDIsIlxccGlfMShVKSJdLFsyLDIsIlxccGlfMShYKSJdLFsyLDAsIlxccGlfMShWKSJdLFs0LDQsIkciXSxbMCwzLCJcXHBpXzEgaV8yIl0sWzAsMSwiXFxwaV8xaV8xIiwyXSxbMSwyLCJcXHBpXzFqXzEiLDJdLFszLDIsIlxccGlfMmpfMiJdLFsyLDAsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFsxLDQsImZfMSIsMix7ImN1cnZlIjoxfV0sWzMsNCwiZl8yIiwwLHsiY3VydmUiOi0xfV0sWzIsNCwiZiIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==

We must show the existence of a unique such that the diagram commutes.

Uniqueness is the easier part to prove: For objects (points), if then ; if then ; and if the assignments agree. For a homotopy path uniqueness follows from a representative . Using a Lebesgue number, may be evenly subdivided into sections either entirely in either or , giving paths where Thus must agree with applying and to each component path, which is clearly invariant under refinement and therefore independent of the precise decomposition.

For existence, we need to show that is independent of the representative . Let by virtue of a homotopy of paths . Once again a Lebesgue number may be used to divide into a grid such that each box is entirely in either or . Assign to the box with bottom-left corner at the paths rightwards along its top and bottom edges respectively, and upwards along its left and right edges respectively. Clearly as paths, and . Since and are constant paths in either or , applying to get paths in for each

where denotes applying or depending on whether a path is in or . It follows from iterations that .

The classical Seifert-Van Kampen theorem concerns the Fundamental group, which can easily be derived from the above theorem. Ronald Brown introduced the groupoid formulation.

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2020, Topology: A categorical approach, §6.7, pp. 139–140 ↩

  2. 2006, Topology and groupoids, §6.7, pp. 240ff ↩