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# 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 are the two most fundamental groups of cells called?

There are only two main types of cells: prokaryotic and eukaryotic. Prokaryotic cells lack a nucleus and other membrane-bound organelles. Eukaryotic cells have a nucleus and other membrane-bound organelles. This allows these cells to have complex functions.

Is a double torus orientable?

A double torus is a topological surface with two holes, formed from the connected sum of two tori. It has an orientable genus of 2 and an Euler characteristic of -2.

### How are fundamental groups calculated?

Van Kampen’s theorem can be used to compute the fundamental group of a space in terms of simpler spaces it is constructed from. If certain conditions are met, the theorem states that for X=⋃Aα, π1(X)=∗απ1(Aα), the free product of the component fundamental groups.

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.

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 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.

Is fundamental group Abelian?

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

Why is the torus orientable?

Orientable surfaces are surfaces for which we can define ‘clockwise’ consistently: thus, the cylinder, sphere and torus are orientable surfaces. In fact, any two-sided surface in space is orientable: thus the disc, cylinder, sphere and n-fold torus, all with or without holes, are orientable surfaces.

### Which is the example of non-orientable surface?

The Möbius strip is a non-orientable surface.

Is the fundamental group Abelian?

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.

## What is the abelianization of the fundamental group?

The abelianization of the fundamental group can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations . Henri Poincaré defined the fundamental group in 1895 in his paper ” Analysis situs “.

Is the fundamental group of a graph abelian?

Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian. For example, the fundamental group of the figure eight is the free group on two letters. More generally, the fundamental group of any graph is a free group.

What is the trivial fundamental group with one element?

Trivial fundamental group. In Euclidean space (Rn) or any convex subset of Rn, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element.

### How to find the fundamental group of an arbitrary topological space?

The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just X . It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space.