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

From Topospaces

## Statement

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

Here, and the identified point is labeled .

Suppose, further that:

- There exists an open subset of containing such that has a strong deformation retraction to .
- There exists an open subset of containing such that has a strong deformation retraction to .

Then, we have the following relationship between the fundamental groups of , , and :

where denotes the free product of groups.