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 is the trivial fundamental group?

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.

Is a torus Contractible?

A meridian γ ⊂ ∂N of the closed solid torus N is an embedded circle which is nullhomotopic in N but not contractible in ∂N (See Definition 2.2).

Is torus A 3d shape?

A 3d shape made by revolving a small circle (radius r) along a line made by a bigger circle (radius R). It usually looks like a ring.

Is torus a shape?

The shape of this ring is called a torus, a donut shape. Nature invented the shape long before our buildings. A torus is the shape of the magnetic field around our bodies, the shape of the magnetic field around Earth. Some physicists think the universe itself is a spinning torus.

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⟩.

Is a circle homotopy equivalent to a point?

To see this, first recall that R2∖{a point} R 2 ∖ { a point } is homotopy equivalent (i.e. it deformation retracts) to a circle.

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 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).

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.

What is the plural of torus?

noun. to·​rus | \ ˈtȯr-əs \ plural tori\ ˈtȯr-​ˌī , -​ˌē \