Functoriality
If f : X → Y is a continuous map, x0 ∈ X and y0 ∈ Y with f(x0) = y0, then every loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting group homomorphism, called the induced homomorphism, is written as π(f) or, more commonly,
This mapping from continuous maps to group homomorphisms is compatible with composition of maps and identity morphisms. In other words, we have a functor from the category of topological spaces with base point to the category of groups.
It turns out that this functor cannot distinguish maps which are homotopic relative to the base point: if f and g : X → Y are continuous maps with f(x0) = g(x0) = y0, and f and g are homotopic relative to {x0}, then f* = g*. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:
As an important special case, if X is path-connected then any two basepoints give isomorphic fundamental groups, with isomorphism given by a choice of path between the given basepoints.
The fundamental group functor takes products to products and coproducts to coproducts. That is, if X and Y are path connected, then
and
(In the latter formula, denotes the wedge sum of topological spaces, and * the free product of groups.) Both formulas generalize to arbitrary products. Furthermore the latter formula is a special case of the Seifert–van Kampen theorem which states that the fundamental group functor takes pushouts along inclusions to pushouts.
Read more about this topic: Fundamental Group