Connected sum of manifolds - Revision history

Vipul: /* Fundamental group */

2011-07-29T00:41:11Z

Fundamental group

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	==Fundamental group==		==Fundamental group==

	~~{{further\|~~[[Fundamental group of connected sum~~]]}}~~		* [[Fundamental group of connected sum is free product of fundamental groups in dimension at least three]]: This fails in dimension two, because the [[circle]] <math>S^1</math> has nontrivial fundamental group.
			* [[Connected sum of simply connected manifolds is simply connected]]
	~~The fundamental group of the connected sum of two manifolds~~ is ~~the~~ free product of ~~their~~ fundamental groups ~~(''this needs to be verified''). In particular~~, the fundamental group ~~of a connected~~ sum of simply connected manifolds is simply connected.

	==Related notions==		==Related notions==

Vipul: /* Fundamental group */

2011-07-28T01:47:16Z

Fundamental group

← Older revision		Revision as of 01:47, 28 July 2011
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	{{further\|[[Fundamental group of connected sum]]}}		{{further\|[[Fundamental group of connected sum]]}}

	The fundamental group of the connected sum of two manifolds is the free product of their fundamental groups. In particular, the fundamental group of a connected sum of simply connected manifolds is simply connected.		The fundamental group of the connected sum of two manifolds is the free product of their fundamental groups (''this needs to be verified''). In particular, the fundamental group of a connected sum of simply connected manifolds is simply connected.

	==Related notions==		==Related notions==

Vipul: /* Definition */

2011-07-28T01:25:43Z

Definition

← Older revision		Revision as of 01:25, 28 July 2011
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	In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together. {{further\|[[homotopy type of connected sum depends on choice of gluing map]]}}		In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together. {{further\|[[homotopy type of connected sum depends on choice of gluing map]]}}


	==Homology==		==Homology==

Vipul at 19:09, 2 April 2011

2011-04-02T19:09:21Z

← Older revision		Revision as of 19:09, 2 April 2011
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	Let <math>M_1</math> and <math>M_2</math> be [[connected manifold]]s. A '''connected sum''' of <math>M_1</math> and <math>M_2</math>, denoted <math>M_1 \# M_2</math>, is constructed as follows. Let <math>f_i:\R^n \to U_i</math> be homeomorphisms where <math>U_i</math> are open subsets of <math>M_i</math>. Let <math>M_i'</math> denote the complement in <math>M_i</math> of the image of the open unit ball in <math>\R^n</math>, under <math>f_i</math>. Then the connected sum is the quotient of <math>M_1 \sqcup M_2</math> under the identification of the boundary <math>S^{n-1}</math>s with each other, via the composite <math>f_2 \circ f_1^{-1}</math>.		Let <math>M_1</math> and <math>M_2</math> be [[connected manifold]]s. A '''connected sum''' of <math>M_1</math> and <math>M_2</math>, denoted <math>M_1 \# M_2</math>, is constructed as follows. Let <math>f_i:\R^n \to U_i</math> be homeomorphisms where <math>U_i</math> are open subsets of <math>M_i</math>. Let <math>M_i'</math> denote the complement in <math>M_i</math> of the image of the open unit ball in <math>\R^n</math>, under <math>f_i</math>. Then the connected sum is the quotient of <math>M_1 \sqcup M_2</math> under the identification of the boundary <math>S^{n-1}</math>s with each other, via the composite <math>f_2 \circ f_1^{-1}</math>.

	In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together.		In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together. {{further\|[[homotopy type of connected sum depends on choice of gluing map]]}}


	==Homology==		==Homology==

Vipul at 18:35, 2 April 2011

2011-04-02T18:35:28Z

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	==Definition==		==Definition==

	Let <math>M_1</math> and <math>M_2</math> be [[connected manifold]]s. A '''connected sum''' of <math>M_1</math> and <math>M_2</math>, denoted <math>M_1 \~~sharp~~ M_2</math>, is constructed as follows. Let <math>f_i:\R^n \to U_i</math> be homeomorphisms where <math>U_i</math> are open subsets of <math>M_i</math>. Let <math>M_i'</math> denote the complement in <math>M_i</math> of the image of the open unit ball in <math>\R^n</math>, under <math>f_i</math>. Then the connected sum is the quotient of <math>M_1 \sqcup M_2</math> under the identification of the boundary <math>S^{n-1}</math>s with each other, via the composite <math>f_2 \circ f_1^{-1}</math>.		Let <math>M_1</math> and <math>M_2</math> be [[connected manifold]]s. A '''connected sum''' of <math>M_1</math> and <math>M_2</math>, denoted <math>M_1 \# M_2</math>, is constructed as follows. Let <math>f_i:\R^n \to U_i</math> be homeomorphisms where <math>U_i</math> are open subsets of <math>M_i</math>. Let <math>M_i'</math> denote the complement in <math>M_i</math> of the image of the open unit ball in <math>\R^n</math>, under <math>f_i</math>. Then the connected sum is the quotient of <math>M_1 \sqcup M_2</math> under the identification of the boundary <math>S^{n-1}</math>s with each other, via the composite <math>f_2 \circ f_1^{-1}</math>.

	In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together.		In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together.

Vipul: 9 revisions

2008-05-11T19:42:23Z

9 revisions

← Older revision	Revision as of 19:42, 11 May 2008
(No difference)

Vipul at 22:02, 13 January 2008

2008-01-13T22:02:05Z

← Older revision		Revision as of 22:02, 13 January 2008
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	The interesting phenomena occur at <math>n</math> and <math>n-1</math>, because this is where the gluing is occurring.		The interesting phenomena occur at <math>n</math> and <math>n-1</math>, because this is where the gluing is occurring.

			==Fundamental group==

			{{further\|[[Fundamental group of connected sum]]}}

			The fundamental group of the connected sum of two manifolds is the free product of their fundamental groups. In particular, the fundamental group of a connected sum of simply connected manifolds is simply connected.

	==Related notions==		==Related notions==

Vipul at 23:20, 13 December 2007

2007-12-13T23:20:40Z

← Older revision		Revision as of 23:20, 13 December 2007
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	The interesting phenomena occur at <math>n</math> and <math>n-1</math>, because this is where the gluing is occurring.		The interesting phenomena occur at <math>n</math> and <math>n-1</math>, because this is where the gluing is occurring.

			==Related notions==

			* [[Fiber sum]]
			* [[Symplectic sum]]
			* [[Knot sum]]

Vipul: /* Homology */

2007-12-03T14:17:20Z

Homology

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 ==Homology==
 The homology of the connected sum can be computed using the [[Mayer-Vietoris homology sequence]] for open sets obtained by ''enlarging'' the <math>M_i'</math>s slightly, and using the fact that <math>M_i'</math> is a [[strong deformation retract]] of <math>M_i</math> minus a point.
 The interesting phenomena occur at <math>n</math> and <math>n-1</math>, because this is where the gluing is occurring.

Vipul at 20:07, 2 December 2007

2007-12-02T20:07:20Z

← Older revision		Revision as of 20:07, 2 December 2007
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	Let <math>M_1</math> and <math>M_2</math> be [[connected manifold]]s. A '''connected sum''' of <math>M_1</math> and <math>M_2</math>, denoted <math>M_1 \sharp M_2</math>, is constructed as follows. Let <math>f_i:\R^n \to U_i</math> be homeomorphisms where <math>U_i</math> are open subsets of <math>M_i</math>. Let <math>M_i'</math> denote the complement in <math>M_i</math> of the image of the open unit ball in <math>\R^n</math>, under <math>f_i</math>. Then the connected sum is the quotient of <math>M_1 \sqcup M_2</math> under the identification of the boundary <math>S^{n-1}</math>s with each other, via the composite <math>f_2 \circ f_1^{-1}</math>.		Let <math>M_1</math> and <math>M_2</math> be [[connected manifold]]s. A '''connected sum''' of <math>M_1</math> and <math>M_2</math>, denoted <math>M_1 \sharp M_2</math>, is constructed as follows. Let <math>f_i:\R^n \to U_i</math> be homeomorphisms where <math>U_i</math> are open subsets of <math>M_i</math>. Let <math>M_i'</math> denote the complement in <math>M_i</math> of the image of the open unit ball in <math>\R^n</math>, under <math>f_i</math>. Then the connected sum is the quotient of <math>M_1 \sqcup M_2</math> under the identification of the boundary <math>S^{n-1}</math>s with each other, via the composite <math>f_2 \circ f_1^{-1}</math>.

			In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together.

	==Homology==		==Homology==
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	<math>\tilde{H}_{n-1}(M_1 \sharp M_2) = \tilde{H}_{n-1}(M_1) \oplus \tilde{H}_{n-1}(M_2)</math>		<math>\tilde{H}_{n-1}(M_1 \sharp M_2) = \tilde{H}_{n-1}(M_1) \oplus \tilde{H}_{n-1}(M_2)</math>

	~~It turns out that the result holds for~~ [[compact connected manifold]]s ~~even if ''one'' of them~~ is non-orientable; this ~~requires a little more argument.~~		If both <math>M_1</math> and <math>M_2</math> are [[compact connected manifold]]s and <math>M_1</math> is non-orientable but <math>M_2</math> is orientable, then the sequence:

			<math>0 \to \tilde{H}_{n-1}(S^{n-1}) \to \tilde{H}_{n-1}(M_1 \setminus p) \to \tilde{H}_{n-1}(M_1) \to 0</math>

			is exact, and this yields, along with Mayer-Vietoris, that:

			<math>\tilde{H}_{n-1}(M_1 \sharp M_2) = \tilde{H}_{n-1}(M_1) \oplus \tilde{H}_{n-1}(M_2)</math>

	If ''both'' are non-orientable, however, then an exceptional situation occurs.		If ''both'' are non-orientable, however, then an exceptional situation occurs.
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	The observations given above yield that when both <math>M_1</math> and <math>M_2</math> are compact connected orientable, then the top homology of their connected sum is again <math>\mathbb{Z}</math>, viz the connected sum is again orientable. This can also be seen directly by the definition of orientability.		The observations given above yield that when both <math>M_1</math> and <math>M_2</math> are compact connected orientable, then the top homology of their connected sum is again <math>\mathbb{Z}</math>, viz the connected sum is again orientable. This can also be seen directly by the definition of orientability.

			===Euler characteristic===

			The Euler characteristics are related by the following formula when both <math>M_1</math> and <math>M_2</math> are [[compact connected manifold]]s:

			<math>\chi(M_1 \sharp M_2) = \chi(M_1) + \chi(M_2) - \chi(S^n)</math>