2-torus

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Definition

As a product space

The 2-torus, sometimes simply called the torus, is defined as the product (equipped with the product topology) of two circles, i.e., it is defined as $S^{1}\times S^{1}$. The 2-torus is also denoted $T^{2}$.

The term torus more generally refers to a product of finitely many copies of the circle, equipped with the product topology. The $n$-torus is sometimes denoted as $T^{n}$.

As a subspace of $\mathbb {R} ^{3}$

A 2-torus in $\mathbb {R} ^{3}$ is obtained as the surface of revolution achieved by revolving a circle about a line in its plane that does not intersect it.

Topological space properties

Property	Satisfied?	Is the property a homotopy-invariant property of topological spaces?	Explanation	Corollary properties satisfied/dissatisfied
manifold	Yes	No	product of manifolds is manifold -- it is a product of two circles. Also, it can be embedded as a closed submanifold in $\mathbb {R} ^{3}$.	satisfies: metrizable space, second-countable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc.
path-connected space	Yes	Yes	path-connectedness is product-closed -- it is a product of circles, which are path-connected spaces.	satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous)
simply connected space	No	Yes	The circle isn't simply connected, and fundamental group of product is product of fundamental groups.	dissatisfies: simply connected manifold
rationally acyclic space	No	Yes	The second homology group is isomorphic to the group of integers, hence it is nontrivial and has nontrivial torsion-free part. See homology of torus	dissatisfies: acyclic space, weakly contractible space, contractible space
space with Euler characteristic zero	Yes	Yes	Euler characteristic of product is product of Euler characteristics, and the circle has Euler characteristic zero. Alternatively, we can use that Euler characteristic of compact connected nontrivial Lie group is zero. See homology of torus.
space with Euler characteristic one	No	Yes	The Euler characteristic is 0, see above.
compact space	Yes	No	product of circles, which are compact; see Tychonoff's theorem	satisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness

Algebraic topology

Homology

Further information: homology of torus

The homology groups with coefficients in $\mathbb {Z}$ are as follows: $H_{0}(S^{1}\times S^{1})\cong \mathbb {Z}$, $H_{1}(S^{1}\times S^{1})\cong \mathbb {Z} \oplus \mathbb {Z}$, and $H_{2}(S^{1}\times S^{1})\cong \mathbb {Z}$. All higher homology groups are zero.