What is the fundamental group of the 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 torus variables?

Torus is a 2-dimensional surface and hence can be parametrized by 2 independent variables which are obviously the 2 angles: α = angle in the x/y-plane, around the z-axis, 0° ≤ α < 360° β = angle around the x/y-plane, 0° ≤ β < 360°

Is a 3 torus flat?

The three-dimensional torus is just one of 10 different flat finite worlds. There are also flat infinite worlds such as the three-dimensional analogue of an infinite cylinder.

Are 3-manifolds classified?

Characteristic classes and characteristic numbers are the corresponding generalized homological invariants, but they do not classify manifolds in higher dimension (they are not a complete set of invariants): for instance, orientable 3-manifolds are parallelizable (Steenrod’s theorem in low-dimensional topology), so all …

Is the human body a torus?

Topologically speaking, a human is a torus. Your digestive system is the hole in the doughnut. Interestingly, this means in a two-dimensional world, an organism couldn’t have a similar structure, since the digestive system would completely separate the animal into two halves.

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 function of a torus?

Torus-bearing pit membranes control water movement between tracheary elements of vascular plants, while at the same time they inhibit spread of air embolisms. They are common in gymnosperms but relatively rare in angiosperms.

How do you get torus?

If you take a ring and circularly trace around with a pencil, you get a torus. In modern design software, it is fairly easy to draw them by using a revolve command with a circle as a cross-section.

How many dimensions is a torus?

In the topological world, a torus is a two-dimensional space, or surface, with one hole. (To be a bit fancier, it is an orientable surface of genus one.)

Is torus a 2 dimensional?

Unless I’m very mistaken, the surface of a torus is 2-dimensional, as is the surface of a sphere. The reason being that being on the surface you can only move in 2 dimensions, up or down is not well defined.

How many 3 manifolds are there?

Amazingly, every compact 2-manifold is homeomorphic to either a sphere (orientable), a connected sum of tori (orientable), or a connected sum of projective planes (nonorientable). There are infinitely many 3-manifolds.

Is a torus a manifold?

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard.