Cohomology of real projective space - Revision history

Vipul: /* Coefficients in a 2-divisible ring */

2011-09-30T21:07:27Z

Coefficients in a 2-divisible ring

← Older revision		Revision as of 21:07, 30 September 2011
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	\|}		\|}

	===Coefficients in a 2-divisible ring===		===Coefficients in a module over a 2-divisible ring===

	Suppose <math>M</math> has the structure of a module over a unital ring <math>R</math> where 2 is invertible. Then, in particular, we know that <math>2M = M</math> and <math>T = 0</math>. Thus, both <math>M/2M</math> and <math>T</math> are equal to <math>0</math>. We get:		Suppose <math>M</math> has the structure of a module over a unital ring <math>R</math> where 2 is invertible. Then, in particular, we know that <math>2M = M</math> and <math>T = 0</math>. Thus, both <math>M/2M</math> and <math>T</math> are equal to <math>0</math>. We get:

Vipul: /* Statement */

2011-09-30T20:32:26Z

Statement

← Older revision		Revision as of 20:32, 30 September 2011
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	space = real projective space}}		space = real projective space}}

	==~~Statement~~==		==Cohomology groups in piecewise form==

	===Odd-dimensional projective space with coefficients in integers===		===Odd-dimensional projective space with coefficients in integers===
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	This in particular applies to the case that <math>M</math> is the group <math>\mathbb{Z}/2\mathbb{Z}</math>, i.e., when we are taking coefficients in the field of two elements.		This in particular applies to the case that <math>M</math> is the group <math>\mathbb{Z}/2\mathbb{Z}</math>, i.e., when we are taking coefficients in the field of two elements.

	==Cohomology groups in tabular form==		==Cohomology groups in tabular form==

Vipul: /* Coefficients in an abelian group */

2011-09-30T20:30:18Z

Coefficients in an abelian group

← Older revision		Revision as of 20:30, 30 September 2011
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	===Coefficients in an abelian group===		===Coefficients in an abelian group===

	We let the abelian group be <math>M</math>. Denote by <math>T</math> the 2-torsion of <math>M</math> and by <math>2M</~~amth~~> the submodule comprising doubles of elements.		We let the abelian group be <math>M</math>. Denote by <math>T</math> the 2-torsion of <math>M</math> and by <math>2M</math> the submodule comprising doubles of elements.

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Cohomology groups with integer coefficients in tabular form */

2011-09-30T20:29:15Z

Cohomology groups with integer coefficients in tabular form

@@ Line 37: / Line 37: @@
 In particular, these results are valid over the field of rational numbers or over any field of characteristic zero, or indeed any characteristic other than 2.
-==Cohomology groups with integer coefficients in tabular form==
+===Coefficients in characteristic two===
 We illustrate how the cohomology groups work for small values of <math>n</math>. Note that for <math>p > n</math>, all cohomology groups <math>H^p</math> are zero, so we omit those cells for visual ease.
 {| class="sortable" border="1"
-! <math>n</math> !! Real projective space <math>\R\mathbb{P}^n</math> !! Orientable? !! <math>H_0</math> !! <math>H_1</math> !! <math>H_2</math> !! <math>H_3</math> !! <math>H_4</math> !! <math>H_5</math>
+! <math>n</math> !! Real projective space <math>\R\mathbb{P}^n</math> !! Orientable? !! <math>H^0</math> !! <math>H^1</math> !! <math>H^2</math> !! <math>H^3</math> !! <math>H^4</math> !! <math>H^5</math>
 |-
 | 1 || [[circle]] || Yes || <math>\mathbb{Z}</math> || <math>\mathbb{Z}</math>
@@ Line 53: / Line 62: @@
 |-
 | 5 || [[RP^5]] || Yes || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}</math>
 |}

Vipul: /* Coefficients in a 2-divisible ring */

2011-09-30T20:19:34Z

Coefficients in a 2-divisible ring

← Older revision		Revision as of 20:19, 30 September 2011
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	Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.		Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

	===Coefficients in a 2-divisible ring===		===Coefficients in a module over a 2-divisible ring===

	If we consider the cohomology with coefficients in a module <math>M</math> over a ring <math>R</math> where 2 is invertible, then we have:		If we consider the cohomology with coefficients in a module <math>M</math> over a ring <math>R</math> where 2 is invertible, then we have:

Vipul: /* Even-dimensional projective space with coefficients in an abelian group */

2011-09-30T20:13:11Z

Even-dimensional projective space with coefficients in an abelian group

← Older revision		Revision as of 20:13, 30 September 2011
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	For an abelian group <math>M</math>, the cohomology is given by:		For an abelian group <math>M</math>, the cohomology is given by:

	<math>H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0\\ M/2M, & \qquad p \ \operatorname{even}, 0 < p \le n\\ T, & p \ \operatorname{odd}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>		<math>H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0\\ M/2M, & \qquad p \ \operatorname{even}, 0 < p \le n\\ T, & \qquad p \ \operatorname{odd}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>

	Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.		Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

Vipul: /* Odd-dimensional projective space with coefficients in an abelian group */

2011-09-30T20:12:37Z

Odd-dimensional projective space with coefficients in an abelian group

← Older revision		Revision as of 20:12, 30 September 2011
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	For an abelian group <math>M</math>, the cohomology is given by:		For an abelian group <math>M</math>, the cohomology is given by:

	<math>H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0,n\\ M/2M, &\qquad p \ \operatorname{even}, 0 < p < n\\ T, & \qquad p \ \operatorname{odd}, 0 < p < n \\, 0 & \qquad \operatorname{otherwise}\end{array}\right.</math>		<math>H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0,n\\ M/2M, &\qquad p \ \operatorname{even}, 0 < p < n\\ T, & \qquad p \ \operatorname{odd}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>

	Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.		Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

Vipul: /* Cohomology groups with integer coefficients in tabular form */

2011-09-30T20:11:54Z

Cohomology groups with integer coefficients in tabular form

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 | 5 || [[RP^5]] || Yes || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}</math>
 |}
 ==Reality checks==

Vipul: /* Facts used */

2011-09-30T20:00:43Z

Facts used

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 |-
 | 5 || [[RP^5]] || Yes || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}</math>
 |}

Vipul: /* Even-dimensional projective space with coefficients in an abelian group */

2011-07-28T01:13:54Z

Even-dimensional projective space with coefficients in an abelian group

← Older revision		Revision as of 01:13, 28 July 2011
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	Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.		Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

			===Coefficients in a 2-divisible ring===

			If we consider the cohomology with coefficients in a module <math>M</math> over a ring <math>R</math> where 2 is invertible, then we have:

			<math>H^p(\mathbb{P}^n(\R);M) := \left\lbrace\begin{array}{rl} M, & p = 0 \\ M, & p = n, n \ \operatorname{odd}\\ 0, & p = n, n \ \operatorname{even}\\ 0, & p \ne 0,n \\\end{array}\right.</math>

			In particular, these results are valid over the field of rational numbers or over any field of characteristic zero, or indeed any characteristic other than 2.

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