Homology of connected sum - Revision history

Vipul at 01:24, 28 July 2011

2011-07-28T01:24:32Z

← Older revision		Revision as of 01:24, 28 July 2011
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	{{effect of operation\|connected sum\|homology groups}}		{{effect of operation\|connected sum\|homology groups}}

	==~~=Homology in low and high dimensions=~~==		==Statement==

	~~In all dimensions other than~~ <math>n</math> and <math>n-1</math>~~, we~~ have ~~the following formula~~:		Suppose <math>M_1</math> and <math>M_2</math> are the [[connected manifold]]s of dimension <math>n</math> whose connected sum is being taken. Assume both We have:

	~~<math>\tilde~~{~~H}_i(M_1 \# M_2)~~ = ~~\tilde{H}_i(M_1) \oplus \tilde{H}_i(M_2)</math>~~		{\| class="sortable" border="1"
			! Case for <math>p</math> !! Additional condition on <math>M_1,M_2</math> !! What is known about <math>H_p(M_1)</math> and <math>H_p(M_2)</math>? !! Formula for <math>H_p(M_1 \# M_2)</math> in terms of homology groups of <math>M_1</math> and <math>M_2</math>
	This does not require any conditions on the manifolds, and only uses the fact that the [[point-deletion inclusion]] (inclusion of manifold minus a point into the manifold) induces isomorphism on all homologies other than <math>~~n,n-1~~</math>.		\|-
			\| 0 \|\| none \|\| both isomorphic to <math>\mathbb{Z}</math> \|\| <math>\mathbb{Z}</math> (because both are connected, so is their connected sum).
	~~===In the second highest dimension===~~		\|-
			\| Greater than 0, less than <math>n - 1</math> \|\| none \|\| both are finitely generated abelian groups \|\| <math>H_p(M_1) \oplus H_p(M_2)</math>
	~~In dimension~~ <math>~~n-1~~</math>~~, we need to know~~ about ~~the nature of the map from~~ <math>~~S^{n-1}~~</math> ~~into~~ <math>~~M_i \setminus p~~</math> ~~as far as~~ <math>(~~n-1~~)~~^{th}~~</math> homology ~~is concerned. Clearly, the inclusion~~ of <math>~~S^{n-1}~~</math> ~~inside~~ <math>M</math> ~~is nullhomotopic, because it factors through a contractible open set.~~		\|-
			\| <math>n - 1</math> \|\| Both manifolds are compact, at least one of them is orientable \|\| both are finitely generated abelian groups, at least one is free abelian \|\| <math>H_{n-1}(M_1) \oplus H_{n-1}(M_2)</math>
	If <math>~~M_i~~</math> ~~is a [[compact connected orientable manifold]] then the inclusion of~~ <math>~~M_1~~ \~~setminus p~~</math> ~~induces isomorphism on the~~ <math>(n-1~~)^{th}~~</math> ~~homology, hence the induced map~~ <math>~~H_{n-1}~~(~~S^{n-1}~~) \~~to H_{~~n-1~~}(M_1')~~</math> ~~is zero. Thus if ''both''~~ manifolds are compact ~~connected~~ orientable, ~~then Mayer-Vietoris yields that:~~		\|-
			\| <math>n-1</math> \|\| other cases \|\| ? \|\| ?
	<math>~~\tilde{H}_{n-1}(M_1 \# M_2) = \tilde{H}_~~{n-1}(M_1) \oplus ~~\tilde{H}_~~{n-1}(M_2)</math>		\|-
			\| <math>n</math> \|\| Both are compact and orientable \|\| both are <math>\mathbb{Z}</math> \|\| <math>\mathbb{Z}</math>
	~~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>n</math> \|\| other cases \|\| ? \|\| ?
	<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>		\|-
			\| Greater than <math>n</math> \|\| none \|\| both are zero groups \|\| 0
	~~is exact, and this yields, along with Mayer~~-~~Vietoris, that:~~		\|}

	<math>~~\tilde{H}_{~~n~~-1}(M_1 \# 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.~~

	~~===In the highest dimension===~~

	~~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===		===Euler characteristic===

Vipul: /* Euler characteristic */

2011-07-27T03:17:22Z

Euler characteristic

← Older revision		Revision as of 03:17, 27 July 2011
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	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:		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>		<math>\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - \chi(S^n)</math>

Vipul at 18:36, 2 April 2011

2011-04-02T18:36:06Z

← Older revision		Revision as of 18:36, 2 April 2011
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	In all dimensions other than <math>n</math> and <math>n-1</math>, we have the following formula:		In all dimensions other than <math>n</math> and <math>n-1</math>, we have the following formula:

	<math>\tilde{H}_i(M_1 \~~sharp~~ M_2) = \tilde{H}_i(M_1) \oplus \tilde{H}_i(M_2)</math>		<math>\tilde{H}_i(M_1 \# M_2) = \tilde{H}_i(M_1) \oplus \tilde{H}_i(M_2)</math>

	This does not require any conditions on the manifolds, and only uses the fact that the [[point-deletion inclusion]] (inclusion of manifold minus a point into the manifold) induces isomorphism on all homologies other than <math>n,n-1</math>.		This does not require any conditions on the manifolds, and only uses the fact that the [[point-deletion inclusion]] (inclusion of manifold minus a point into the manifold) induces isomorphism on all homologies other than <math>n,n-1</math>.
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	If <math>M_i</math> is a [[compact connected orientable manifold]] then the inclusion of <math>M_1 \setminus p</math> induces isomorphism on the <math>(n-1)^{th}</math> homology, hence the induced map <math>H_{n-1}(S^{n-1}) \to H_{n-1}(M_1')</math> is zero. Thus if ''both'' manifolds are compact connected orientable, then Mayer-Vietoris yields that:		If <math>M_i</math> is a [[compact connected orientable manifold]] then the inclusion of <math>M_1 \setminus p</math> induces isomorphism on the <math>(n-1)^{th}</math> homology, hence the induced map <math>H_{n-1}(S^{n-1}) \to H_{n-1}(M_1')</math> is zero. Thus if ''both'' manifolds are compact connected orientable, then Mayer-Vietoris yields 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>		<math>\tilde{H}_{n-1}(M_1 \# M_2) = \tilde{H}_{n-1}(M_1) \oplus \tilde{H}_{n-1}(M_2)</math>

	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:		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:
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	is exact, and this yields, along with Mayer-Vietoris, that:		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>		<math>\tilde{H}_{n-1}(M_1 \# 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.

Vipul: 2 revisions

2008-05-11T19:46:18Z

2 revisions

← Older revision	Revision as of 19:46, 11 May 2008
(No difference)

Vipul at 14:18, 3 December 2007

2007-12-03T14:18:15Z

← Older revision		Revision as of 14:18, 3 December 2007
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			{{effect of operation\|connected sum\|homology groups}}

	===Homology in low and high dimensions===		===Homology in low and high dimensions===

Vipul at 14:17, 3 December 2007

2007-12-03T14:17:15Z

New page

===Homology in low and high dimensions===

In all dimensions other than <math>n</math> and <math>n-1</math>, we have the following formula:

<math>\tilde{H}_i(M_1 \sharp M_2) = \tilde{H}_i(M_1) \oplus \tilde{H}_i(M_2)</math>

This does not require any conditions on the manifolds, and only uses the fact that the [[point-deletion inclusion]] (inclusion of manifold minus a point into the manifold) induces isomorphism on all homologies other than <math>n,n-1</math>.

===In the second highest dimension===

In dimension <math>n-1</math>, we need to know about the nature of the map from <math>S^{n-1}</math> into <math>M_i \setminus p</math> as far as <math>(n-1)^{th}</math> homology is concerned. Clearly, the inclusion of <math>S^{n-1}</math> inside <math>M</math> is nullhomotopic, because it factors through a contractible open set.

If <math>M_i</math> is a [[compact connected orientable manifold]] then the inclusion of <math>M_1 \setminus p</math> induces isomorphism on the <math>(n-1)^{th}</math> homology, hence the induced map <math>H_{n-1}(S^{n-1}) \to H_{n-1}(M_1')</math> is zero. Thus if ''both'' manifolds are compact connected orientable, then Mayer-Vietoris yields 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 <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.

===In the highest dimension===

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>