Homology of spheres: Difference between revisions

Latest revision as of 22:51, 9 January 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 group and the topological space/family is sphere
Get more specific information about sphere | Get more computations of homology group

In this article, we briefly describe the homology groups of spheres, a proof using the Mayer-Vietoris homology sequence, and explanations in terms of cellular and simplicial homology.

Statement

Reduced version over integers

For $n$ a nonnegative integer, we have the following result for the reduced homology groups:

${\tilde{H}}_{k} (S^{n}) = 0, k \neq n$ :

and:

${\tilde{H}}_{n} (S^{n}) ≅ Z$

Unreduced version over integers

We need to make cases based on whether $n = 0$ or $n$ is a positive integer:

$n = 0$ case: $H_{0} (S^{0}) ≅ Z \oplus Z$ and $H_{k} (S^{0})$ is trivial for $k > 0$ .
$n > 0$ case: $H_{0} (S^{n}) ≅ H_{n} (S^{n}) ≅ Z$ and $H_{k} (S^{n})$ is trivial for $k \neq 0, n$ .

Reduced version over a module $M$ over a ring $R$

For $n$ a nonnegative integer, we have the following result for the reduced homology groups:

${\tilde{H}}_{k} (S^{n}) = 0, k \neq n$ :

and:

${\tilde{H}}_{n} (S^{n}) ≅ M$

Unreduced version over a module $M$ over a ring $R$

We need to make cases based on whether $n = 0$ or $n$ is a positive integer:

$n = 0$ case: $H_{0} (S^{0}) ≅ M \oplus M$ and $H_{k} (S^{0})$ is trivial for $k > 0$ .
$n > 0$ case: $H_{0} (S^{n}) ≅ H_{n} (S^{n}) ≅ M$ and $H_{k} (S^{n})$ is trivial for $k \neq 0, n$ .

Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant	General description	Description of value for spheres
Betti numbers	The $k^{t h}$ Betti number $b_{k}$ is the rank of the $k^{t h}$ homology group.	For $n = 0$ , $b_{0} = 2$ , all other $b_{k}$ are $0$ ; for $n > 0$ , $b_{0} = b_{n} = 1$ , all other $b_{k}$ s are $0$ .
Poincare polynomial	Generating polynomial for Betti numbers	$2$ for $n = 0$ , $1 + x^{n}$ for $n > 0$
Euler characteristic	$\sum_{k = 0}^{\infty} (- 1)^{k} b_{k}$	$2$ for $n$ even, $0$ for $n$ odd.

Facts used

Proof using singular homology

Equivalence of reduced and unreduced version

The equivalence follows from the fact that reduced and unreduced homology groups coincide for $k > 0$ and for $k = 0$ , the unreduced homology group is obtained from the reduced one by adding a copy of $Z$ (or, if working over another ring or module, the base ring or module).

Proof of reduced version

The case $n = 0$ is clear: the space $S^{0}$ is a discrete two-point space, hence it has two single-point path components, so the zeroth homology group is $M^{2 - 1} = M^{1} = M$ . Higher homology groups are trivial because the cycle and boundary groups both coincide with the group of all functions to $S^{0}$ , so the homology group is trivial.

In general, we use induction, starting with the base case $n = 0$ . The inductive step follows from fact (1) and the fact that each $S^{n}$ is the suspension of $S^{n - 1}$ .

Proof using cellular homology

See (2): CW structure of spheres.

Proof using simplicial homology

See (3): Simplicial structure of spheres.

@@ Line 1: / Line 1: @@
-In this article, we briefly describe how to compute the homology groups of spheres using the [[Mayer-Vietoris homology sequence]].
+{{homotopy invariant computation|
+invariant = homology group|
+space = sphere}}
-{{fillin}}
+In this article, we briefly describe the [[computation of::homology group]]s of [[specific information about::sphere]]s, a proof using the [[Mayer-Vietoris homology sequence]], and explanations in terms of cellular and simplicial homology.
+==Statement==
+===Reduced version over integers===
+For <math>n</math> a nonnegative integer, we have the following result for the [[reduced homology group]]s:
+<math>\tilde{H}_k(S^n) = 0, k \ne n</math>:
+and:
+<math>\tilde{H}_n(S^n) \cong \mathbb{Z}</math>
+===Unreduced version over integers===
+We need to make cases based on whether <math>n = 0</math> or <math>n</math> is a positive integer:
+* <math>n = 0</math> case: <math>H_0(S^0) \cong \mathbb{Z} \oplus \mathbb{Z}</math> and <math>H_k(S^0)</math> is trivial for <math>k > 0</math>.
+* <math>n > 0</math> case: <math>H_0(S^n) \cong H_n(S^n) \cong \mathbb{Z}</math> and <math>H_k(S^n)</math> is trivial for <math>k \ne 0,n</math>.
+===Reduced version over a module <math>M</math> over a ring <math>R</math>===
+For <math>n</math> a nonnegative integer, we have the following result for the [[reduced homology group]]s:
+<math>\tilde{H}_k(S^n) = 0, k \ne n</math>:
+and:
+<math>\tilde{H}_n(S^n) \cong M</math>
+===Unreduced version over a module <math>M</math> over a ring <math>R</math>===
+We need to make cases based on whether <math>n = 0</math> or <math>n</math> is a positive integer:
+* <math>n = 0</math> case: <math>H_0(S^0) \cong M \oplus M</math> and <math>H_k(S^0)</math> is trivial for <math>k > 0</math>.
+* <math>n > 0</math> case: <math>H_0(S^n) \cong H_n(S^n) \cong M</math> and <math>H_k(S^n)</math> is trivial for <math>k \ne 0,n</math>.
+==Related invariants==
+These are all invariants that can be computed in terms of the homology groups.
+{| class="sortable" border="1"
+! Invariant !! General description !! Description of value for spheres
+|-
+| [[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. || For <math>n = 0</math>, <math>b_0 = 2</math>, all other <math>b_k</math> are <math>0</math>; for <math>n > 0</math>, <math>b_0 = b_n = 1</math>, all other <math>b_k</math>s are <math>0</math>.
+|-
+| [[Poincare polynomial]] || Generating polynomial for Betti numbers || <math>2</math> for <math>n = 0</math>, <math>1 + x^n</math> for <math>n > 0</math>
+|-
+| [[Euler characteristic]] || <math>\sum_{k=0}^\infty (-1)^k b_k</math> || <math>2</math> for <matH>n</math> even, <math>0</math> for <math>n</math> odd.
+|}
+==Facts used==
+# [[uses::Homology for suspension]]
+# [[uses::CW structure of spheres]]
+# [[uses::Simplicial structure of spheres]]
+==Proof using singular homology==
+===Equivalence of reduced and unreduced version===
+The equivalence follows from the fact that reduced and unreduced homology groups coincide for <math>k > 0</math> and for <math>k = 0</math>, the unreduced homology group is obtained from the reduced one by adding a copy of <math>\mathbb{Z}</math> (or, if working over another ring or module, the base ring or module).
+===Proof of reduced version===
+The case <math>n = 0</math> is clear: the space <math>S^0</math> is a discrete two-point space, hence it has two single-point path components, so the zeroth homology group is <math>M^{2 - 1} = M^1 = M</math>. Higher homology groups are trivial because the cycle and boundary groups both coincide with the group of ''all'' functions to <math>S^0</math>, so the homology group is trivial.
+In general, we use induction, starting with the base case <math>n = 0</math>. The inductive step follows from fact (1) and the fact that each <math>S^n</math> is the suspension of <math>S^{n-1}</math>.
+==Proof using cellular homology==
+See (2): [[CW structure of spheres]].
+==Proof using simplicial homology==
+See (3): [[Simplicial structure of spheres]].