Fundamental group of wedge sum relative to basepoints with neighborhoods that deformation retract to them is free product of fundamental groups

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Statement

Suppose (X,x_0) and (Y,y_0) are based topological spaces. Suppose, further, that X and Y are both path-connected spaces (otherwise, we basically care only about the path components of X and Y). Consider the Wedge sum (?):

\! (Z,z_0) = (X,x_0) \vee (Y,y_0)

Here, Z = (X \sqcup Y)/\{ x_0, y_0 \} and the identified point \! x_0 \sim y_0 is labeled z_0.

Suppose, further that:

  • There exists an open subset A of X containing x_0 such that A has a strong deformation retraction to x_0.
  • There exists an open subset B of Y containing y_0 such that B has a strong deformation retraction to y_0.

Then, we have the following relationship between the fundamental groups of (X,x_0), (Y,y_0), and (Z,z_0):

\! \pi_1(Z,z_0) \cong \pi_1(X,x_0) * \pi_1(Y,y_0)

where \! * denotes the free product of groups.

Related facts

Similar facts

Facts used

  1. Seifert-van Kampen theorem