Sphere: Difference between revisions

Revision as of 04:00, 9 November 2010

Definition

As a subset of Euclidean space

The unit $n$ -sphere $S^{n}$ is defined as the subset of Euclidean space $R^{n + 1}$ comprising those points whose distance from the origin is $1$ .

$S^{n} = {(x_{0}, x_{1}, \dots, x_{n}) ∣ x_{0}^{2} + x_{1}^{2} + \dots + x_{n}^{2} = 1}$

Particular cases

$n$	sphere $S^{n}$
0	$S^{0}$ -- discrete two-point space
1	circle
2	2-sphere
3	3-sphere

Equivalent spaces

Space	How strongly is it equivalent to the circle?
boundary of the $(n + 1)$ -hypercube	homeomorphic; not diffeomorphic because of sharp edges
boundary of the $(n + 1)$ -simplex	homeomorphic; not diffeomorphic because of sharp edges
ellipsoid in $R^{n + 1}$	equivalent via affine transformation

Algebraic topology

Homology groups

With coefficients in $Z$ , the $n$ -sphere $S^{n}$ has $H_{n} (S^{n}) = Z$ and $H_{k} (S^{n}) = 0$ for $k \neq n$ . In particular, the $n$ -sphere is $(n - 1)$ -connected.

Interpretations in terms of various homology theories:

Fill this in later

With coefficients in any $R$ -module $M$ for a ring $R$ , the $n$ -sphere $S^{n}$ has $H_{n} (S^{n}) = M$ and $H_{k} (S^{n}) = 0$ for all $k \neq n$ .

Cohomology groups and cohomology ring

With coefficients in $Z$ , the $n$ -sphere $S^{n}$ has $H^{n} (S^{n}) = Z$ and $H^{k} (S^{n}) = 0$ for $k \neq n$ . In particular, the $n$ -sphere is $(n - 1)$ -connected.

With coefficients in any $R$ -module $M$ for a ring $R$ , the $n$ -sphere $S^{n}$ has $H^{n} (S^{n}) = M$ and $H^{k} (S^{n}) = 0$ for all $k \neq n$ .

@@ Line 20: / Line 20: @@
 | 3 || [[3-sphere]]
 |}
+==Equivalent spaces==
+{| class="sortable" border="1"
+! Space !! How strongly is it equivalent to the circle?
+|-
+| boundary of the <math>(n+1)</math>-hypercube || homeomorphic; not diffeomorphic because of sharp edges
+|-
+| boundary of the <math>(n+1)</math>-simplex || homeomorphic; not diffeomorphic because of sharp edges
+|-
+| ellipsoid in <math>\R^{n+1}</math> || equivalent via affine transformation
+|}
+==Algebraic topology==
+===Homology groups===
+With coefficients in <math>\mathbb{Z}</math>, the <math>n</math>-sphere <math>S^n</math> has <math>H_n(S^n) = \mathbb{Z}</math> and <math>H_k(S^n) = 0</math> for <math>k \ne n</math>. In particular, the <math>n</math>-sphere is <math>(n - 1)</math>-connected.
+Interpretations in terms of various homology theories:
+{{fillin}}
+With coefficients in any <math>R</math>-module <math>M</math> for a ring <math>R</math>, the <math>n</math>-sphere <math>S^n</math> has <math>H_n(S^n) = M</math> and <math>H_k(S^n) = 0</math> for all <math>k \ne n</math>.
+===Cohomology groups and cohomology ring===
+With coefficients in <math>\mathbb{Z}</math>, the <math>n</math>-sphere <math>S^n</math> has <math>H^n(S^n) = \mathbb{Z}</math> and <math>H^k(S^n) = 0</math> for <math>k \ne n</math>. In particular, the <math>n</math>-sphere is <math>(n - 1)</math>-connected.
+With coefficients in any <math>R</math>-module <math>M</math> for a ring <math>R</math>, the <math>n</math>-sphere <math>S^n</math> has <math>H^n(S^n) = M</math> and <math>H^k(S^n) = 0</math> for all <math>k \ne n</math>.