Homology of compact non-orientable surfaces - Revision history

Vipul: /* Related invariants */

2011-07-28T02:15:32Z

Related invariants

← Older revision		Revision as of 02:15, 28 July 2011
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	<math>H_k(P_n;M) = \lbrace\begin{array}{rl} M, & k = 0 \\ M^{n-1}, & k = 1\\0, & k \ge 2 \\\end{array}</math>		<math>H_k(P_n;M) = \lbrace\begin{array}{rl} M, & k = 0 \\ M^{n-1}, & k = 1\\0, & k \ge 2 \\\end{array}</math>

			==Homology groups with integer coefficients in tabular form==

			We illustrate how the homology groups work for small values of <math>n</math>. Note that <math>H_2</math> is zero and all higher <math>H_p</math> are zero.

			{\| class="sortable" border="1"
			! <math>n</math> !! <math>P_n</math> !! <math>H_0(P_n)</math> !! <math>H_1(P_n)</math> !! <math>H_2(P_n)</math>
			\|-
			\| 1 \|\| [[real projective plane]] \|\| <math>\mathbb{Z}</math> \|\| <math>\mathbb{Z}/2\mathbb{Z}</math> \|\| 0
			\|-
			\| 2 \|\| [[Klein bottle]] \|\| <math>\mathbb{Z}</math> \|\| <math>\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> \|\| 0
			\|-
			\| 3 \|\| [[Dyck's surface]] \|\| <math>\mathbb{Z}</math> \|\| <math>\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> \|\| 0
			\|}

	==Related invariants==		==Related invariants==

Vipul: /* Related invariants */

2011-04-02T20:03:01Z

Related invariants

← Older revision		Revision as of 20:03, 2 April 2011
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	! Invariant !! General description !! Description of value for connected sum of <math>n</math> copies of real projective plane !! Comment		! Invariant !! General description !! Description of value for connected sum of <math>n</math> copies of real projective plane !! Comment
	\|-		\|-
	\| [[Betti number]]s \|\| The <math>k^{th}</math> Betti number <math>b_k</math> is the free ~~rank~~ of the <math>k^{th}</math> homology group. \|\| <math>b_0 = 1</math>, <math>b_1 = n - 1</math>, all higher <math>b_k</math> are zero \|\|		\| [[Betti number]]s \|\| The <math>k^{th}</math> Betti number <math>b_k</math> is the rank of the torsion-free part of the <math>k^{th}</math> homology group. \|\| <math>b_0 = 1</math>, <math>b_1 = n - 1</math>, all higher <math>b_k</math> are zero \|\|
	\|-		\|-
	\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + (n - 1)x</math>\|\|		\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + (n - 1)x</math>\|\|

Vipul: /* Related invariants */

2011-04-02T20:02:44Z

Related invariants

← Older revision		Revision as of 20:02, 2 April 2011
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	! Invariant !! General description !! Description of value for connected sum of <math>n</math> copies of real projective plane !! Comment		! Invariant !! General description !! Description of value for connected sum of <math>n</math> copies of real projective plane !! 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_0 = 1</math>, <math>b_1 = n - 1</math>, all higher <math>b_k</math> are zero \|\|		\| [[Betti number]]s \|\| The <math>k^{th}</math> Betti number <math>b_k</math> is the free rank of the <math>k^{th}</math> homology group. \|\| <math>b_0 = 1</math>, <math>b_1 = n - 1</math>, all higher <math>b_k</math> are zero \|\|
	\|-		\|-
	\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + (n - 1)x</math>\|\|		\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + (n - 1)x</math>\|\|

Vipul at 20:02, 2 April 2011

2011-04-02T20:02:12Z

← Older revision		Revision as of 20:02, 2 April 2011
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	===Unreduced version over a module===		===Unreduced version over a module===

	{{~~fillin~~}} ~~-- basically the behavior is governed by the behavior for the homology of the [[real projective plane]]~~, ~~see [[homology of real projective space]].~~		If we consider the homology with coefficients in a module <math>M</math> over a ring <math>R</math> where 2 is invertible, then we have:

			<math>H_k(P_n;M) = \lbrace\begin{array}{rl} M, & k = 0 \\ M^{n-1}, & k = 1\\0, & k \ge 2 \\\end{array}</math>

	==Related invariants==		==Related invariants==

Vipul at 19:59, 2 April 2011

2011-04-02T19:59:24Z

← Older revision		Revision as of 19:59, 2 April 2011
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	\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + (n - 1)x</math>\|\|		\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + (n - 1)x</math>\|\|
	\|-		\|-
	\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math> \|\| <math>2 - n</math> \|\| In particular, this means that the Euler characteristic is negative for <math>n > 2</math>.		\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math> \|\| <math>2 - n</math> \|\| In particular, this means that the Euler characteristic is negative for <math>n > 2</math>. Note that if the Euler characteristic of a compact surface is odd and at most <math>1</math>, then the surface ''must'' be non-orientable and its homeomorphism type can be computed (using <math>2 - n = \chi</math>. If the Euler characteristic is even and at most <math>0</math>, then there is a unique possibility for a compact orientable surface and a unique possibility for a compact non-orientable surface. For an Euler characteristic of <math>2</math>, there is a unique compact orientable surface and no compact non-orientable surface. For an Euler characteristic bigger than <math>2</math>, there is no (connected) compact surface possible.
	\|}		\|}

Vipul: /* Related invariants */

2011-04-02T19:55:50Z

Related invariants

← Older revision		Revision as of 19:55, 2 April 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Invariant !! General description !! Description of value for ~~torus~~ !! Comment		! Invariant !! General description !! Description of value for connected sum of <math>n</math> copies of real projective plane !! 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_0 = 1</math>, <math>b_1 = n - 1</math>, all higher <math>b_k</math> are zero \|\|		\| [[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_0 = 1</math>, <math>b_1 = n - 1</math>, all higher <math>b_k</math> are zero \|\|

Vipul at 19:55, 2 April 2011

2011-04-02T19:55:28Z

← Older revision		Revision as of 19:55, 2 April 2011
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	<math>H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ \mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1\\ 0, & k \ge 2 \\\end{array}</math>		<math>H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ \mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1\\ 0, & k \ge 2 \\\end{array}</math>

	===Reduced version over the ~~inte~~		===Reduced version over the integers===

			We have:

			<math>\tilde{H}_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} 0, & k = 0 \\ \mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1\\ 0, & k \ge 2 \\\end{array}</math>

			===Unreduced version over a module===

			{{fillin}} -- basically the behavior is governed by the behavior for the homology of the [[real projective plane]], see [[homology of real projective space]].

			==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 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_0 = 1</math>, <math>b_1 = n - 1</math>, all higher <math>b_k</math> are zero \|\|
			\|-
			\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + (n - 1)x</math>\|\|
			\|-
			\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math> \|\| <math>2 - n</math> \|\| In particular, this means that the Euler characteristic is negative for <math>n > 2</math>.
			\|}

Vipul at 19:52, 2 April 2011

2011-04-02T19:52:12Z

← Older revision		Revision as of 19:52, 2 April 2011
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	We have:		We have:

	<math>H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ \mathbb{Z}^{(n-1~~)/2~~} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1~~, n \ \operatorname{odd} \\ \mathbb{Z}^{(n-2)/2} \oplus \mathbb{Z}/2\mathbb{Z}, & n \ \operatorname{odd}~~ \\ 0, & k \ge 2 \\\end{array}</math>		<math>H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ \mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1\\ 0, & k \ge 2 \\\end{array}</math>

			===Reduced version over the inte

Vipul at 19:51, 2 April 2011

2011-04-02T19:51:05Z

← Older revision		Revision as of 19:51, 2 April 2011
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	We have:		We have:

	<math>H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ ?, & k = 1 \\ 0, & k \ge 2 \\\end{array}</math>		<math>H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ \mathbb{Z}^{(n-1)/2} \oplus \mathbb{Z}/2\mathbb{Z}, & k = 1, n \ \operatorname{odd} \\ \mathbb{Z}^{(n-2)/2} \oplus \mathbb{Z}/2\mathbb{Z}, & n \ \operatorname{odd} \\ 0, & k \ge 2 \\\end{array}</math>

Vipul: Created page with "{{homotopy invariant computation| invariant = homology| space = compact non-orientable surface}} ==Statement== Suppose $k$ is a positive integer. We denote by $..."$

2011-04-02T19:48:21Z

Created page with "{{homotopy invariant computation| invariant = homology| space = compact non-orientable surface}} ==Statement== Suppose <math>k</math> is a positive integer. We denote by <math>..."

New page

{{homotopy invariant computation|
invariant = homology|
space = compact non-orientable surface}}

==Statement==

Suppose <math>k</math> is a positive integer. We denote by <math>P_n</math> (''not standard notation, should try to find something'') the [[connected sum of manifolds|connected sum]] of the [[real projective plane]] with itself <math>n</math> times, i.e., the connected sum of <math>n</math> copies of the real projective plane.

===Unreduced version over the integers===

We have:

<math>H_k(P_n;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & k = 0 \\ ?, & k = 1 \\ 0, & k \ge 2 \\\end{array}</math>

Homology of compact non-orientable surfaces - Revision history

Vipul: /* Related invariants */

Vipul: /* Related invariants */

Vipul: /* Related invariants */

Vipul at 20:02, 2 April 2011

Vipul at 19:59, 2 April 2011

Vipul: /* Related invariants */

Vipul at 19:55, 2 April 2011

Vipul at 19:52, 2 April 2011

Vipul at 19:51, 2 April 2011

Vipul: Created page with "{{homotopy invariant computation| invariant = homology| space = compact non-orientable surface}} ==Statement== Suppose k is a positive integer. We denote by ..."

Vipul: Created page with "{{homotopy invariant computation| invariant = homology| space = compact non-orientable surface}} ==Statement== Suppose $k$ is a positive integer. We denote by $..."$