Wedge of two circles: Difference between revisions

Revision as of 17:24, 30 December 2010

Definition

Abstract 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 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 $S^{1} \lor S^{1}$ .

Definition as a subset of Euclidean space

The 'wedge of two circles can be viewed concretely as the following subset of the Euclidean plane:

${(x, y) ∣ [(x - 1)^{2} + y^{2} - 1] [(x + 1)^{2} + y^{2} - 1] = 0}$

Here, the first factor corresponds to the circle with center $(1, 0)$ and radius $1$ , and the second factor corresponds to the circle with center $(- 1, 0)$ and radius $1$ . The two circles are tangent to each other at $(0, 0)$ , which is the point where the circles are wedged together.

Equivalent spaces

Space	How strongly is it equivalent to the wedge of two circles?
configuration space with parameter $(2, 2)$ , i.e., the space obtained by removing two points from the plane.	Homotopy-equivalent
quotient space of disjoint union of two squares by the identification of one side of each.	Homotopy-equivalent

Algebraic topology

Homology groups

The homology groups with coefficients in $Z$ are as follows: $H_{0} (S^{1} \lor S^{1}) ≅ Z$ , $H_{1} (S^{1} \lor S^{1}) ≅ Z \oplus Z$ , and all higher homology groups are zero.

More generally, the homology groups with coefficients in any module $M$ over a commutative unital ring $R$ are as follows: $H_{0} (S^{1} \lor S^{1}; M) ≅ M$ , $H_{1} (S^{1} \lor S^{1}; M) ≅ M \oplus M$ , 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).

Cohomology groups

The cohomology groups with coefficients in $Z$ are as follows: $H^{0} (S^{1} \lor S^{1}) ≅ Z$ , $H^{1} (S^{1} \lor S^{1}) ≅ Z \oplus Z$ , and all higher cohomology groups are zero.

More generally, the cohomology groups with coefficients in a commutative unital ring $R$ are as follows: $H_{0} (S^{1} \lor S^{1}; R) ≅ R$ , $H^{(} S^{1} \lor S^{1}; R) ≅ R \oplus R$ , and all higher cohomology groups are zero. The cohomology group is $R [x, y] / (x^{2}, y^{2}, x y)$ where $x$ and $y$ are a basis for the free module $R \oplus R$ over $R$ .

Homotopy groups

The set of path components $π_{0} (S^{1} \lor S^{1})$ is the one-point set. The fundamental group $π_{1} (S^{1} \lor S^{1})$ 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).

@@ Line 1: / Line 1: @@
 ==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''' (sometimes also called the '''figure of eight''') is defined as the [[defining ingredient::wedge sum]] of two [[defining ingredient::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>.
+===Definition as a subset of Euclidean space===
+The ''''wedge of two circles''' can be viewed concretely as the following subset of the Euclidean plane:
+<math>\! \{ (x,y) \mid [(x - 1)^2 + y^2 - 1][(x + 1)^2 + y^2 - 1] = 0 \}</math>
+Here, the first factor corresponds to the circle with center <math>(1,0)</math> and radius <math>1</math>, and the second factor corresponds to the circle with center <math>(-1,0)</math> and radius <math>1</math>. The two circles are tangent to each other at <math>(0,0)</math>, which is the point where the circles are ''wedged'' together.
+==Equivalent spaces==
+{| class="sortable" border="1"
+! 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
+|-
+| quotient space of disjoint union of two squares by the identification of one side of each. || Homotopy-equivalent
+|}
+==Algebraic topology==
+===Homology groups===
+The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(S^1 \vee S^1) \cong \mathbb{Z}</math>,<math>H_1(S^1 \vee S^1) \cong \mathbb{Z} \oplus \mathbb{Z}</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]]).
+===Cohomology groups===
+The cohomology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H^0(S^1 \vee S^1) \cong \mathbb{Z}</math>,<math>H^1(S^1 \vee S^1) \cong \mathbb{Z} \oplus \mathbb{Z}</math>, and all higher cohomology groups are zero.
+More generally, the cohomology groups with coefficients in a commutative unital ring <math>R</math> are as follows: <math>H_0(S^1 \vee S^1;R) \cong R</math>, <math>H^(S^1 \vee S^1;R) \cong R \oplus R</math>, and all higher cohomology groups are zero. The cohomology group is <math>R[x,y]/(x^2,y^2,xy)</math> where <math>x</math> and <math>y</math> are a basis for the free module <math>R \oplus R</math> over <math>R</math>.
+===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]]).