In an 1894 paper, he introduced the concept of the fundamental group.

The fundamental group measures the 1-dimensional hole structure of a space.

Instead, the flag is defined only up to the action of the fundamental group.

On the connection between the fundamental groups of some related spaces.

For example, the fundamental group of the figure eight is the free group on two letters.

Thus the fundamental group of is equal to for any .

However, they need not have isomorphic fundamental groups (at a particular base point).

Their fundamental groups are isomorphic because each space is simply connected.

He made important contributions to topology, especially to the study of fundamental groups.

So these three-manifolds are completely determined by their fundamental group.

As such, each side can thus an element of the fundamental group .

There are several ways to define the orbifold fundamental group.

For example, the fundamental group "counts" how many paths in the space are essentially different.

This has many implications for the structure of the fundamental group:

See fundamental group for more on this type of induced homomorphism.

The fundamental group of a rose is free, with one generator for each petal.

The fundamental group is the first and simplest of the homotopy groups.

More generally, the fundamental group of any graph is a free group.

The fundamental group of the presentation complex is the group G itself.

The maps on fundamental groups are given as follows.

It is clear from this that the fundamental group of is trivial.

A useful example is the induced homomorphism of fundamental groups.

This allows calculations such as the fundamental group of a symmetric square.

A path-connected space with a trivial fundamental group is said to be simply connected.

The fundamental group is generated by loops winding around each arc.