Wedge of two circles - Revision history

Vipul: /* Algebraic topology */

2010-12-30T18:13:59Z

Algebraic topology

← Older revision		Revision as of 18:13, 30 December 2010
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	More generally, the homology groups with coefficients in any module <math>M</math> over a commutative unital ring <math>R</math> are as follows: <math>H_0(S^1 \vee S^1;M) \cong M</math>, <math>H_1(S^1 \vee S^1;M) \cong M \oplus M</math>, and all higher homology groups are zero.		More generally, the homology groups with coefficients in any module <math>M</math> over a commutative unital ring <math>R</math> are as follows: <math>H_0(S^1 \vee S^1;M) \cong M</math>, <math>H_1(S^1 \vee S^1;M) \cong M \oplus M</math>, and all higher homology groups are zero.

	These results follows from the [[homology computation for spheres]] (specifically, the circle case) and [[reduced homology of wedge sum relative to basepoints with ~~contractible~~ neighborhoods is direct sum of reduced homologies]] (which in turn follows from the [[Mayer-Vietoris homology sequence]]).		These results follows from the [[homology computation for spheres]] (specifically, the circle case) and [[reduced homology of wedge sum relative to basepoints with neighborhoods that deformation retract to them is direct sum of reduced homologies]] (which in turn follows from the [[Mayer-Vietoris homology sequence]]).

	===Cohomology groups===		===Cohomology groups===
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	===Homotopy groups===		===Homotopy groups===

	The [[set of path components]] <math>\pi_0(S^1 \vee S^1)</math> is the one-point set. The [[fundamental group]] <math>\pi_1(S^1 \vee S^1)</math> is isomorphic to [[free group:F2]] -- this follows from the fact that [[fundamental group of wedge sum relative to basepoints with ~~contractible~~ neighborhoods is free product of fundamental groups]] (which in turn follows from the [[Seifert-van Kampen theorem]]).		The [[set of path components]] <math>\pi_0(S^1 \vee S^1)</math> is the one-point set. The [[fundamental group]] <math>\pi_1(S^1 \vee S^1)</math> is isomorphic to [[free group:F2]] -- this follows from the fact that [[fundamental group of wedge sum relative to basepoints with neighborhoods that deformation retract to them is free product of fundamental groups]] (which in turn follows from the [[Seifert-van Kampen theorem]]).

Vipul: /* Equivalent spaces */

2010-12-30T17:25:56Z

Equivalent spaces

← Older revision		Revision as of 17:25, 30 December 2010
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	! Space !! How strongly is it equivalent to the wedge of two circles?		! Space !! How strongly is it equivalent to the wedge of two circles?
	\|-		\|-
	\| ~~[[configuration space]] with parameter <math>(2,2)</math>, i.e., the~~ space obtained by removing two points from the plane. \|\| Homotopy-equivalent		\| The space obtained by removing two points from the Euclidean plane. \|\| Homotopy-equivalent
			\|-
			\| The space obtained by removing three points from the [[2-sphere]] \|\| Homotopy-equivalent
	\|-		\|-
	\| quotient space of disjoint union of two squares by the identification of one side of each. \|\| Homotopy-equivalent		\| quotient space of disjoint union of two squares by the identification of one side of each. \|\| Homotopy-equivalent

Vipul at 17:24, 30 December 2010

2010-12-30T17:24:47Z

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 ==Definition==
-The '''wedge of two circles''' (sometimes also called the '''figure of eight''') is defined as the [[wedge sum]] of two [[circle]]s with respect to any basepoint in both. Note that since a circle is a [[homogeneous space]], the choice of basepoint does not affect the homeomorphism type of the wedge.
+===Abstract definition===
 The wedge of two circles is denoted <math>S^1 \vee S^1</math>.

Vipul: Created page with '==Definition== The '''wedge of two circles''' (sometimes also called the '''figure of eight''') is defined as the wedge sum of two circles with respect to any basepoint ...'

2010-12-30T16:56:54Z

Created page with '==Definition== The '''wedge of two circles''' (sometimes also called the '''figure of eight''') is defined as the wedge sum of two circles with respect to any basepoint ...'

New page

==Definition==

The '''wedge of two circles''' (sometimes also called the '''figure of eight''') is defined as the [[wedge sum]] of two [[circle]]s with respect to any basepoint in both. Note that since a circle is a [[homogeneous space]], the choice of basepoint does not affect the homeomorphism type of the wedge.

The wedge of two circles is denoted <math>S^1 \vee S^1</math>.