Vipul at 00:39, 31 March 2011

2011-03-31T00:39:32Z

← Older revision		Revision as of 00:39, 31 March 2011
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	{{further\|[[Groupprops:Group cohomology of cyclic group:Z2]] (on the Group Properties Wiki)}}		{{further\|[[Groupprops:Group cohomology of cyclic group:Z2]] (on the Group Properties Wiki)}}

	This topological space is a [[path-connected space\|path-connected]] [[aspherical space]] and its [[fundamental group]] is [[cyclic group:Z2\|cyclic of order two]]. This follows from the definition in terms of the countable-dimensional sphere (making the countable-dimensional sphere its double cover) and the fact that the [[countable-dimensional sphere is contractible]]		This topological space is a [[path-connected space\|path-connected]] [[aspherical space]] and its [[fundamental group]] is [[groupprops:cyclic group:Z2\|cyclic of order two]]. This follows from the definition in terms of the countable-dimensional sphere (making the countable-dimensional sphere its double cover) and the fact that the [[countable-dimensional sphere is contractible]]

	Thus, it can be viewed as a [[classifying space]] for [[cyclic group:Z2]]. In particular, this means that the ''topological'' homology and cohomology groups of this space are the same as the group homology and cohomology groups of [[cyclic group:Z2]].		Thus, it can be viewed as a [[classifying space]] for [[groupprops:cyclic group:Z2\|the cyclic group of order two]]. In particular, this means that the ''topological'' homology and cohomology groups of this space are the same as the group homology and cohomology groups of [[cyclic group:Z2]].

	==Algebraic topology==		==Algebraic topology==
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	===Homotopy===		===Homotopy===

	The zeroth homotopy is a one-point set, the first homotopy group is [[cyclic group:Z2]], and all higher homotopy groups are zero (i.e., the space is an [[aspherical space]]). This follows because the universal cover, the [[countable-dimensional sphere]], is a double cover and also the fact that [[countable-dimensional sphere is contractible]].		The zeroth homotopy is a one-point set, the first homotopy group is [[groupprops:cyclic group:Z2\|cyclic of order two]], and all higher homotopy groups are zero (i.e., the space is an [[aspherical space]]). This follows because the universal cover, the [[countable-dimensional sphere]], is a double cover and also the fact that [[countable-dimensional sphere is contractible]].

Vipul: Created page with "{{particular topospace}} ==Definition== ===As a projective space=== This topological space is the projective space corresponding to a countable-dimensional vector space over t..."

2011-03-31T00:38:09Z

Created page with "{{particular topospace}} ==Definition== ===As a projective space=== This topological space is the projective space corresponding to a countable-dimensional vector space over t..."

New page

{{particular topospace}}

==Definition==

===As a projective space===

This topological space is the projective space corresponding to a countable-dimensional vector space over the real numbers. Explicitly, denote by <math>\R^\infty</math> the space of all sequences of real numbers with at most finitely many nonzero entries, where the addition and scalar multiplication are coordinate-wise. This is a countable-dimensional vector space over <math>\R</math>. We thus have an action of the multiplicative group <math>\R^*</math> on it by scalar multiplication. The quotient of <math>\R^\infty \setminus \{ 0 \}</math> by this action, given the [[quotient topology]], is defined as the '''countable-dimensional real projective space''' and is denoted as <math>\R\mathbb{P}^\omega</math> or <math>\R\mathbb{P}^\infty</math>.

===As a quotient of a countable-dimensional sphere by antipode identification===

Define <math>S^\infty</math>, the [[countable-dimensional sphere]], as follows: denote by <math>\R^\infty</math> the space of all sequences of real numbers with at most finitely many nonzero entries, where the addition and scalar multiplication are coordinate-wise. <math>S^\infty</math> is the subset of <math>\R^\infty</math> comprising those sequences whose sum of squares of entries is <math>1</math>.

<math>\R\mathbb{P}^\infty</math> is defined as the quotient of <math>S^\infty</math> by the following equivalence relation: any element is identified with its ''antipode'', i.e., the element obtained by taking negatives of all the entries. This quotient is equipped with the [[quotient topology]].

==As a classifying space==

{{further|[[Groupprops:Group cohomology of cyclic group:Z2]] (on the Group Properties Wiki)}}

This topological space is a [[path-connected space|path-connected]] [[aspherical space]] and its [[fundamental group]] is [[cyclic group:Z2|cyclic of order two]]. This follows from the definition in terms of the countable-dimensional sphere (making the countable-dimensional sphere its double cover) and the fact that the [[countable-dimensional sphere is contractible]]

Thus, it can be viewed as a [[classifying space]] for [[cyclic group:Z2]]. In particular, this means that the ''topological'' homology and cohomology groups of this space are the same as the group homology and cohomology groups of [[cyclic group:Z2]].

==Algebraic topology==

===Homology===

{{further|[[homology of countable-dimensional real projective space]]}}

===Cohomology===

{{further|[[cohomology of countable-dimensional real projective space]]}}

===Homotopy===

The zeroth homotopy is a one-point set, the first homotopy group is [[cyclic group:Z2]], and all higher homotopy groups are zero (i.e., the space is an [[aspherical space]]). This follows because the universal cover, the [[countable-dimensional sphere]], is a double cover and also the fact that [[countable-dimensional sphere is contractible]].

Countable-dimensional real projective space - Revision history

Vipul at 00:39, 31 March 2011

Vipul: Created page with "{{particular topospace}} ==Definition== ===As a projective space=== This topological space is the projective space corresponding to a countable-dimensional vector space over t..."