Is the fundamental group a functor?

Is the fundamental group a functor?

Assigning the fundamental group to a topological space is definitely a functor. But you have to keep in mind that a fundamental group is always taken with respect to a base point, and hence the functor assigns a pair (X,x0) consisting of a topological space X and a point x0∈X to its fundamental group π1(X,x0).

What is the fundamental group of a torus?

The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n.

Are fundamental groups always Abelian?

A Hopf space is a space in which the proof given above for the statement that the fundamental group of a topological group is Abelian still works. Thus, by definition, the fundamental group of a Hopf space is Abelian.

What is the fundamental group of the Klein bottle?

The fundamental group of the Klein bottle can be determined as the group of deck transformations of the universal cover and has the presentation ⟨a, b | ab = b−1a⟩.

What is the trivial fundamental group?

there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. More generally, any star domain and, yet more generally any contractible space has a trivial fundamental group. Thus, the fundamental group does not distinguish between such spaces.

Is every group a fundamental group?

Every group can be realized as the fundamental group of a connected CW-complex of dimension 2 (or higher). As noted above, though, only free groups can occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs).

Is a Klein bottle a torus?

The pinched torus is perhaps the simplest parametrization of the klein bottle in both three and four dimensions. It’s a torus that, in three dimensions, flattens and passes through itself on one side.

Is a torus Simply Connected?

A torus is not simply connected. Neither of the colored loops can be contracted to a point without leaving the surface.

Which groups are fundamental groups?

The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups.

Can a Möbius strip exist?

The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858, though similar structures can be seen in Roman mosaics c. 200–250 AD.