Homology of torus: Difference between revisions

Latest revision as of 02:28, 26 July 2011

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

Statement

We denote by $T^{n}$ the $n$ -dimensional torus, which is the topological space:

$(S^{1})^{n} ≅ S^{1} \times S^{1} \times \dots S^{1}$

i.e., the product of $S^{1}$ , the circle, with itself $n$ times. It is equipped with the product topology. If we think of $S^{1}$ as a group, $T^{n}$ gets a group structure too as the external direct product.

Unreduced version over integers

The $k^{t h}$ homology group $H_{k} (T^{n})$ is a free abelian group $Z^{(\binom{n}{k})}$ of rank $(\binom{n}{k})$ , where $(\binom{n}{k})$ denotes the binomial coefficient, or the number of subsets of size $k$ in a set of size $n$ . In particular, $(\binom{n}{k})$ is positive for $k \in {0, 1, 2, \dots, n}$ and zero for other $k$ . Thus, $H_{k} (T^{n})$ is nontrivial for $0 \leq k \leq n$ and zero for $k > n$ .

Reduced version over integers

The $k^{t h}$ reduced homology group ${\tilde{H}}_{k} (T^{n})$ is a free abelian group of rank $(\binom{n}{k})$ for $k > 0$ and is trivial for $k = 0$ . In particular, it is a nontrivial group for $k \in {1, 2, \dots, n}$ and is zero for other $k$ .

Unreduced version over an abelian group

The $k^{t h}$ homology group $H_{k} (T^{n}; M)$ is a direct sum $M^{(\binom{n}{k})}$ of rank $(\binom{n}{k})$ , where $(\binom{n}{k})$ denotes the binomial coefficient, or the number of subsets of size $k$ in a set of size $n$ . 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 $M$ 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 $n$ and $k$ . Each row corresponds to a value of $n$ and each column corresponds to a value of $k$ for which we are computing $H_{k} (T^{n})$ . If a cell value reads 2, for instance, that means that the corresponding homology group with coefficients in the integers is $Z^{2} = Z \oplus Z$ and the corresponding homology group with coefficients in a module $M^{2}$ .</math> Note that the cell values for $k > n$ are omitted, because all these values are zero:

$n$	Torus $T^{n}$	$H_{0}$ rank	$H_{1}$ rank	$H_{2}$ rank	$H_{3}$ rank	$H_{4}$ rank	$H_{5}$ rank
0	one-point space	1
1	circle	1	1
2	2-torus	1	2	1
3	3-torus	1	3	3	1
4	4-torus	1	4	6	4	1
5	5-torus	1	5	10	10	5	1

Related invariants

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 $k^{t h}$ Betti number $b_{k}$ is the rank of the $k^{t h}$ homology group.	$b_{k} = (\binom{n}{k})$
Poincare polynomial	Generating polynomial for Betti numbers	$(1 + x)^{n}$
Euler characteristic	$\sum_{k = 0}^{\infty} (- 1)^{k} b_{k}$	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).

@@ Line 5: / Line 5: @@
 ==Statement==
-We denote by <math>T^n</math> the <math>k</math>-dimensional torus, which is the topological space:
+We denote by <math>T^n</math> the <math>n</math>-dimensional torus, which is the topological space:
 <math>(S^1)^n \cong S^1 \times S^1 \times \dots S^1</math>
@@ Line 14: / Line 14: @@
 The <math>k^{th}</math> homology group <math>H_k(T^n)</math> is a [[free abelian group]] <math>\mathbb{Z}^{\binom{n}{k}}</math> of rank <math>\binom{n}{k}</math>, where <math>\binom{n}{k}</math> denotes the binomial coefficient, or the number of subsets of size <math>k</math> in a set of size <math>n</math>. In particular, <math>\binom{n}{k}</math> is positive for <math>k \in \{ 0,1,2,\dots,n \}</math> and zero for other <math>k</math>. Thus, <math>H_k(T^n)</math> is nontrivial for <math>0 \le k \le n</math> and zero for <math>k > n</math>.
-Below are given the ranks of homology groups for small values of <math>n</math> and <math>k</math>. Each row corresponds to a value of <math>n</math> and each column corresponds to a value of <math>k</math>. If a cell value reads 2, for instance, that means that the corresponding homology group is <math>\mathbb{Z}^2 = \mathbb{Z} \oplus \mathbb{Z}</math>:
-{| class=sortable" border="1"
-! <math>n,k</math> !! 0 !! 1 !! 2 !! 3 !! 4
-|-
-| 0 || 1 || 0 || 0 || 0 || 0
-|-
-| 1 || 1 || 1 || 0 || 0 || 0
-|-
-| 2 || 1 || 2 || 1 || 0 || 0
-|-
-| 3 || 1 || 3 || 3 || 1 || 0
-|-
-| 4 || 1 || 4 || 6 || 4 || 1
-|}
 ===Reduced version over integers===
@@ Line 42: / Line 26: @@
 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 <math>n</math> and <math>k</math>. Each row corresponds to a value of <math>n</math> and each column corresponds to a value of <math>k</math> for which we are computing <math>H_k(T^n)</math>. If a cell value reads 2, for instance, that means that the corresponding homology group with coefficients in the integers is <math>\mathbb{Z}^2 = \mathbb{Z} \oplus \mathbb{Z}</math> and the corresponding homology group with coefficients in a module <math>M^2</math>.</math> Note that the cell values for <math>k > n</math> are omitted, because all these values are zero:
+{| class=sortable" border="1"
+! <math>n</math> !! Torus <math>T^n</math> !! <math>H_0</math> rank !! <math>H_1</math> rank !! <math>H_2</math> rank !! <math>H_3</math> rank !! <math>H_4</math> rank !! <math>H_5</math> rank
+|-
+| 0 || [[one-point space]] || 1
+|-
+| 1 || [[circle]] || 1 || 1
+|-
+| 2 || [[2-torus]] || 1 || 2 || 1
+|-
+| 3 || [[3-torus]] || 1 || 3 || 3 || 1
+|-
+| 4 || [[4-torus]] || 1 || 4 || 6 || 4 || 1
+|-
+| 5 || [[5-torus]] || 1 || 5 || 10 || 10 || 5 || 1
+|}
 ==Related invariants==
@@ Line 48: / Line 52: @@
 {| class="sortable" border="1"
-! Invariant !! General description !! Description of value for torus
+! Invariant !! General description !! Description of value for torus !! Comment
 |-
-| [[Betti number]]s || The <math>k^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. || <math>b_k = \binom{n}{k}</math>
+| [[Betti number]]s || The <math>k^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. || <math>b_k = \binom{n}{k}</math> ||
 |-
-| [[Poincare polynomial]] || Generating polynomial for Betti numbers || <math>(1 + x)^n</math>
+| [[Poincare polynomial]] || Generating polynomial for Betti numbers || <math>(1 + x)^n</math> ||
 |-
-| [[Euler characteristic]] || <math>\sum_{k=0}^\infty (-1)^k b_k</math> || 0 -- can also be seen from the fact that we have a group.
+| [[Euler characteristic]] || <math>\sum_{k=0}^\infty (-1)^k b_k</math> || 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]]).
 |}
 ==Automorphisms, endomorphisms, and homomorphisms==