What is the fundamental group of a torus?

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.

What is the fundamental group of a torus with one point removed?

A torus with one point removed deformation retracts onto a figure eight, namely the union of two generating circles. More generally, a surface of genus g with one point removed deformation retracts onto a rose with 2g petals, namely the boundary of a fundamental polygon.

What group is a knot?

The knot group of the unknot is the infinite cyclic group C. 3 ∪ {∞}, the z-axis together with infinity is a circle, and the image of an unknot.

What is the fundamental group of the Klein bottle?

The universal cover of both the torus and the Klein bottle is the plane R2. 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⟩.

Why is the fundamental group of the torus Abelian?

Gluing the rectancle to make a torus, this shows that going first around through the hole and then along the outside is homeomorphic to going first along the outside and then through the hole. Since these two path generate the fundamental group of the torus this proves that this group is abelan.

Is the fundamental group Abelian?

The fundamental group is abelian iff basepoint-change homomorphisms depend only on the endpoints.

What topology is in a circle?

A ring topology is a network configuration where device connections create a circular data path. Each networked device is connected to two others, like points on a circle.

Is the Hawaiian earring path connected?

This means that the space is not semi-locally simply connected. Viewed in terms of general topology, it would be hard to sell the earring space as a genuinely “pathological space”: as it is a compact, Hausdorff, connected and locally path-connected metric space, etc.

Is the universe a Klein bottle?

The cover image for Endless Universe is a “double-Klein bottle” decorated with images representing a big bang and cosmic evolution in each “bulb.” The Klein bottle is a hypothetical surface whose inside connects to its outside and back to its inside again.

What is fundamental group give example?

Loosely speaking, the fundamental group measures “the number of holes” in a space. For example, the fundamental group of a point or a line or a plane is trivial, while the fundamental group of a circle is Z.