2-torus - Revision history

Vipul: /* Homology groups */

2015-05-15T06:12:40Z

Homology groups

← Older revision		Revision as of 06:12, 15 May 2015
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	<math>H_p(S^1 \times S^1;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z} & \qquad p = 0,2 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}</math>		<math>H_p(S^1 \times S^1;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z} & \qquad p = 0,2 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}</math>

	The ~~cohomology~~ groups with coefficients in a module <math>M</math> are as follows:		The homology groups with coefficients in a module <math>M</math> are as follows:

	<math>H_p(S^1 \times S^1;M) = \lbrace\begin{array}{rl} M & \qquad p = 0,2 \\ M \oplus M, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}</math>		<math>H_p(S^1 \times S^1;M) = \lbrace\begin{array}{rl} M & \qquad p = 0,2 \\ M \oplus M, & \qquad p = 1\\0, & \qquad p \ge 3\\\end{array}</math>

Vipul: /* Manifold-type invariants */

2011-07-24T16:29:42Z

Manifold-type invariants

← Older revision		Revision as of 16:29, 24 July 2011
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	==Manifold-type invariants==		==Manifold-type invariants==

	~~===Dimension invariants===~~

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
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	\|-		\|-
	\| smallest dimension of Euclidean space in which it can be embedded as a flat submanifold \|\| 4 \|\| Consider the usual embedding of <math>S^1</math> in <math>\R^2</math> as a unit circle. This is an embedding as a flat submanifold. Consider <math>\R^4</math> as the product <math>\R^2 \times \R^2</math> and consider the induced embedding of <math>S^1 \times S^1</math> in it.		\| smallest dimension of Euclidean space in which it can be embedded as a flat submanifold \|\| 4 \|\| Consider the usual embedding of <math>S^1</math> in <math>\R^2</math> as a unit circle. This is an embedding as a flat submanifold. Consider <math>\R^4</math> as the product <math>\R^2 \times \R^2</math> and consider the induced embedding of <math>S^1 \times S^1</math> in it.
			\|-
			\| minimum number of charts needed in an atlas for this manifold \|\| 3 \|\| Pick a point <math>p \in S^1</math>. Then, the subsets <math>S^1 \times (S^1 \setminus \{ p \})</math>, <math>(S^1 \setminus \{ p \}) \times S^1</math>, and a small open neighborhood of the point <math>(p,p)</math>, together form a collection of Euclidean open neighborhoods that cover the manifold.
	\|}		\|}

Vipul at 15:16, 24 July 2011

2011-07-24T15:16:52Z

@@ Line 85: / Line 85: @@
 |-
 | <math>k \ge 2</math> || higher homotopy group || trivial group
 |}

Vipul: /* Homology-based invariants */

2011-07-24T01:58:45Z

Homology-based invariants

← Older revision		Revision as of 01:58, 24 July 2011
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	\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>(1 + x)^n</math> \|\| <math>(1 + x)^2 = 1 + 2x + x^2</math>		\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>(1 + x)^n</math> \|\| <math>(1 + x)^2 = 1 + 2x + x^2</math>
	\|-		\|-
	\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math>, equal to the Poincare polynomial evaluated at -1. \|\| 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]]). \|\| 0		\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math>, equal to the Poincare polynomial evaluated at -1. \|\| 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]]). \|\| 0
	\|}		\|}

Vipul: /* Topological space properties */

2011-07-24T01:49:18Z

Topological space properties

← Older revision		Revision as of 01:49, 24 July 2011
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	\| [[dissatisfies property::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: [[dissatisfies property::acyclic space]], [[dissatisfies property::weakly contractible space]], [[dissatisfies property::contractible space]]		\| [[dissatisfies property::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: [[dissatisfies property::acyclic space]], [[dissatisfies property::weakly contractible space]], [[dissatisfies property::contractible space]]
	\|-		\|-
	\| [[satisfies property::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]].		\| [[satisfies property::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]]. \|\|
	\|-		\|-
	\| [[dissatisfies property::space with Euler characteristic one]] \|\| No \|\| Yes \|\| The Euler characteristic is 0, see above. \|\|		\| [[dissatisfies property::space with Euler characteristic one]] \|\| No \|\| Yes \|\| The Euler characteristic is 0, see above. \|\|

Vipul: /* Algebraic topology */

2011-07-24T01:48:35Z

Algebraic topology

@@ Line 35: / Line 35: @@
 ==Algebraic topology==
-===Homology===
+===Homology groups===
 {{further|[[homology of torus]]}}
-The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(S^1 \times S^1) \cong \mathbb{Z}</math>, <math>H_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z}</math>, and <math>H_2(S^1 \times S^1) \cong \mathbb{Z}</math>. All higher homology groups are zero.
+The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows:

Vipul at 01:26, 22 July 2011

2011-07-22T01:26:32Z

@@ Line 12: / Line 12: @@
 A 2-torus in <math>\R^3</math> is obtained as the surface of revolution achieved by revolving a circle about a line in its plane that does not intersect it.
 ==Algebraic topology==
@@ Line 19: / Line 39: @@
 {{further|[[homology of torus]]}}
-The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(S^1 \times S^1) \cong \mathbb{Z}</math>, <math>H_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z}</math>, and <math>H_2(S^1 \times S^1) \cong \mathbb{Z}</math>.
+The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(S^1 \times S^1) \cong \mathbb{Z}</math>, <math>H_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z}</math>, and <math>H_2(S^1 \times S^1) \cong \mathbb{Z}</math>. All higher homology groups are zero.

Vipul: /* Definition */

2011-07-22T01:22:23Z

Definition

← Older revision		Revision as of 01:22, 22 July 2011
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	===As a product space===		===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 <math>S^1 \times S^1</math>.		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 <math>S^1 \times S^1</math>. The 2-torus is also denoted <math>T^2</math>.

	The term [[torus]] more generally refers to a product of finitely many copies of the circle, equipped with the product topology.		The term [[torus]] more generally refers to a product of finitely many copies of the circle, equipped with the product topology. The <math>n</math>-torus is sometimes denoted as <math>T^n</math>.

	===As a subspace of <math>\R^3</math>===		===As a subspace of <math>\R^3</math>===

Vipul at 00:36, 22 July 2011

2011-07-22T00:36:29Z

← Older revision		Revision as of 00:36, 22 July 2011
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	{{particular topospace}}		{{particular topospace}}
			[[dimension::2\| ]]
	==Definition==		==Definition==

Vipul: Created page with '{{particular topospace}} ==Definition== ===As a product space=== The 2-torus, sometimes simply called the '''torus''', is defined as the product (equipped with the [[product t...'

2010-12-31T19:04:06Z

Created page with '{{particular topospace}} ==Definition== ===As a product space=== The 2-torus, sometimes simply called the '''torus''', is defined as the product (equipped with the [[product t...'

New page

{{particular topospace}}

==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 <math>S^1 \times S^1</math>.

The term [[torus]] more generally refers to a product of finitely many copies of the circle, equipped with the product topology.

===As a subspace of <math>\R^3</math>===

A 2-torus in <math>\R^3</math> is obtained as the surface of revolution achieved by revolving a circle about a line in its plane that does not intersect it.

==Algebraic topology==

===Homology===

{{further|[[homology of torus]]}}

The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(S^1 \times S^1) \cong \mathbb{Z}</math>, <math>H_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z}</math>, and <math>H_2(S^1 \times S^1) \cong \mathbb{Z}</math>.