Homology of countable-dimensional real projective space - Revision history

Vipul: /* Statement */

2011-09-14T20:23:17Z

Statement

← Older revision		Revision as of 20:23, 14 September 2011
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	The homology groups with coefficients in the ring of integers <math>\mathbb{Z}</math> are given as follows:		The homology groups with coefficients in the ring of integers <math>\mathbb{Z}</math> are given as follows:

	<math>H_p(\mathbb{P}^n(\R);\mathbb{Z}) = \lbrace\begin{array}{rl}\mathbb{Z}/2\mathbb{Z}, &p = 1,3,5,\dots\\0, & p = 2,4,6, \dots \\ \mathbb{Z},& p = 0\\\end{array}</math>		<math>H_p(\mathbb{P}^\infty(\R);\mathbb{Z}) = \lbrace\begin{array}{rl}\mathbb{Z}/2\mathbb{Z}, &p = 1,3,5,\dots\\0, & p = 2,4,6, \dots \\ \mathbb{Z},& p = 0\\\end{array}</math>

	===Over an abelian group or module <math>M</math>===		===Over an abelian group or module <math>M</math>===
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	The homology groups with coefficients in a module <math>M</math> over a ring <math>R</math> are given by:		The homology groups with coefficients in a module <math>M</math> over a ring <math>R</math> are given by:

	<math>H_p(\mathbb{P}^n(\R);M) = \lbrace\begin{array}{rl} M/2M, & p=1,3,5,\dots\\ T, & p = 2,4,6, \dots \\ M, & p = 0\\\end{array}</math>		<math>H_p(\mathbb{P}^\infty(\R);M) = \lbrace\begin{array}{rl} M/2M, & p=1,3,5,\dots\\ T, & p = 2,4,6, \dots \\ M, & p = 0\\\end{array}</math>

	where <math>T</math> is the 2-torsion submodule of <math>M</math>, i.e., the submodule of <math>M</math> comprising elements whose double is zero.		where <math>T</math> is the 2-torsion submodule of <math>M</math>, i.e., the submodule of <math>M</math> comprising elements whose double is zero.

Vipul at 16:26, 21 July 2011

2011-07-21T16:26:21Z

← Older revision		Revision as of 16:26, 21 July 2011
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	Note that the third case, where <math>M</math> is 2-divisible but not necessarily uniquely so, cannot arise if <math>M = R</math> and it is a ''unital'' ring. So when taking coefficients over a unital ring, there is no need to distinguish between 2-divisibility and unique 2-divisibility.		Note that the third case, where <math>M</math> is 2-divisible but not necessarily uniquely so, cannot arise if <math>M = R</math> and it is a ''unital'' ring. So when taking coefficients over a unital ring, there is no need to distinguish between 2-divisibility and unique 2-divisibility.

			==Proof==

			The chain complex arising from a CW structure is as follows:

			<math>\dots \stackrel{\cdot 0}{\to} \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z} \to \dots \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z}</math>

			where the subscript for the last written entry is <math>0</math>, and hence the multiplication by 2 maps arise from even to odd subscripts and the multiplication by zero maps arise from odd to even subscripts.

			===Homology computation over integers===

			{\| class="sortable" border="1"
			! Case for <math>p</math> !! Fragment of relevance in chain complex (<math>(p+1)^{th}</math> to <math>p^{th}</math> to <math>(p-1)^{th}</math>) !! Cycle group (kernel from <math>p^{th}</math> to <math>(p-1)^{th}</math> !! Boundary group (image from <math>(p+1)^{th}</math> group to <math>p^{th}</math> group !! Homology group = cycle group/boundary group
			\|-
			\| 0 \|\| <math>\mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z} \stackrel{0}{\to} 0</math> \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math>
			\|-
			\| odd \|\| <math>\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z}</math> \|\| <math>\mathbb{Z}</math> \|\| <math>2\mathbb{Z}</math> \|\| <math>\mathbb{Z}/2\mathbb{Z}</math>
			\|-
			\| even, positive \|\| <math>\mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z}</math> \|\| 0 \|\| 0 \|\| 0
			\|}

			===Homology computation over an abelian group or module <math>M</math>===

			The chain complex remains the same, but each <math>\mathbb{Z}</math> is replaced by <math>M</math>.

			Denote by <math>T</math> the 2-torsion submodule of <math>M</math> and by <math>M/2M</math> the quotient of <math>M</math> by the submodule <math>2M</math> comprising the doubles of elements.

			{\| class="sortable" border="1"
			! Case for <math>p</math> !! Fragment of relevance in chain complex (<math>(p+1)^{th}</math> to <math>p^{th}</math> to <math>(p-1)^{th}</math>) !! Cycle group (kernel from <math>p^{th}</math> to <math>(p-1)^{th}</math> !! Boundary group (image from <math>(p+1)^{th}</math> group to <math>p^{th}</math> group !! Homology group = cycle group/boundary group
			\|-
			\| 0 \|\| <math>M \stackrel{\cdot 0}{\to} M \stackrel{0}{\to} 0</math> \|\| <math>M</math> \|\| 0 \|\| <math>M</math>
			\|-
			\| odd \|\| <math>M \stackrel{\cdot 2}{\to} M \stackrel{\cdot 0}{\to} \mathbb{Z}</math> \|\| <math>M</math> \|\| <math>2M</math> \|\| <math>M/2M</math>
			\|-
			\| even, positive \|\| <math>M \stackrel{\cdot 0}{\to} M \stackrel{\cdot 2}{\to} M</math> \|\| <math>T</math> \|\| 0 \|\| <math>T</math>
			\|}

Vipul: Created page with "{{homotopy invariant computation| invariant = homology group| space = countable-dimensional real projective space}} ==Statement== ===Over the integers=== The homology groups w..."

2011-07-21T16:01:03Z

Created page with "{{homotopy invariant computation| invariant = homology group| space = countable-dimensional real projective space}} ==Statement== ===Over the integers=== The homology groups w..."

New page

{{homotopy invariant computation|
invariant = homology group|
space = countable-dimensional real projective space}}

==Statement==

===Over the integers===

The homology groups with coefficients in the ring of integers <math>\mathbb{Z}</math> are given as follows:

<math>H_p(\mathbb{P}^n(\R);\mathbb{Z}) = \lbrace\begin{array}{rl}\mathbb{Z}/2\mathbb{Z}, &p = 1,3,5,\dots\\0, & p = 2,4,6, \dots \\ \mathbb{Z},& p = 0\\\end{array}</math>

===Over an abelian group or module <math>M</math>===

The homology groups with coefficients in a module <math>M</math> over a ring <math>R</math> are given by:

<math>H_p(\mathbb{P}^n(\R);M) = \lbrace\begin{array}{rl} M/2M, & p=1,3,5,\dots\\ T, & p = 2,4,6, \dots \\ M, & p = 0\\\end{array}</math>

where <math>T</math> is the 2-torsion submodule of <math>M</math>, i.e., the submodule of <math>M</math> comprising elements whose double is zero.

In particular, we see the following cases:

{| class="sortable" border="1"
! Case on <math>R</math> or <math>M</math> !! Conclusion about odd-indexed homology groups, i.e., <math>H_p, p = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H_p, p = 2,4,6,\dots</math>
|-
| <math>M</math> is uniquely 2-divisible, i.e., every element of <math>M</math> has a unique half. This includes the case that <math>M</math> is a field of characteristic not 2. || all zero groups || all zero groups
|-
| <math>M</math> is 2-torsion-free, i.e., no nonzero element of <math>M</math> doubles to zero || unclear || all zero groups
|-
| <math>M</math> is 2-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || all zero groups || unclear
|-
| <math>M = \mathbb{Z}/2^n\mathbb{Z}</math>, <math>n</math> any natural number || all isomorphic to <math>\mathbb{Z}/2\mathbb{Z}</math> || all isomorphic to <math>\mathbb{Z}/2\mathbb{Z}</math>
|-
| <math>M</math> is a finite abelian group || all isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^r</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math> || all isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^r</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math>
|}

Note that the third case, where <math>M</math> is 2-divisible but not necessarily uniquely so, cannot arise if <math>M = R</math> and it is a ''unital'' ring. So when taking coefficients over a unital ring, there is no need to distinguish between 2-divisibility and unique 2-divisibility.