Homology of torus
This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology and the topological space/family is torus
Get more specific information about torus | Get more computations of homology
We denote by the -dimensional torus, which is the topological space:
Unreduced version over integers
The homology group is a free abelian group of rank , where denotes the binomial coefficient, or the number of subsets of size in a set of size . In particular, is positive for and zero for other . Thus, is nontrivial for and zero for .
Reduced version over integers
The reduced homology group is a free abelian group of rank for and is trivial for . In particular, it is a nontrivial group for and is zero for other .
Unreduced version over an abelian group
The homology group is a direct sum of rank , where denotes the binomial coefficient, or the number of subsets of size in a set of size . The behavior is qualitatively the same as over the integers. Note that this result is the same regardless of whether we think of the homology with coefficients in as an abelian group or as a module over some other commutative unital ring.
Reduced version over an abelian group
This is the same as the unreduced version, except that the zeroth homology group is zero.
Homology groups in tabular form
Below are given the ranks of homology groups for small values of and . Each row corresponds to a value of and each column corresponds to a value of for which we are computing . If a cell value reads 2, for instance, that means that the corresponding homology group with coefficients in the integers is and the corresponding homology group with coefficients in a module .</math> Note that the cell values for are omitted, because all these values are zero:
These are all invariants that can be computed in terms of the homology groups.
|Invariant||General description||Description of value for torus||Comment|
|Betti numbers||The Betti number is the rank of the homology group.|
|Poincare polynomial||Generating polynomial for Betti numbers|
|Euler characteristic||0 (hence it is a space with Euler characteristic zero)||This can also be seen from the fact that we have a group (see Euler characteristic of compact connected nontrivial Lie group is zero) or from the fact that it is a product of circles, and the Euler characteristic of the circle is zero (see Euler characteristic of product is product of Euler characteristics).|