Homology of connected sum: Difference between revisions

Latest revision as of 01:24, 28 July 2011

This article describes the effect of the connected sum operation on the following invariant: homology groups

Statement

Suppose $M_{1}$ and $M_{2}$ are the connected manifolds of dimension $n$ whose connected sum is being taken. Assume both We have:

Case for $p$	Additional condition on $M_{1},M_{2}$	What is known about $H_{p}(M_{1})$ and $H_{p}(M_{2})$ ?	Formula for $H_{p}(M_{1}\#M_{2})$ in terms of homology groups of $M_{1}$ and $M_{2}$
0	none	both isomorphic to $\mathbb {Z}$	$\mathbb {Z}$ (because both are connected, so is their connected sum).
Greater than 0, less than $n-1$	none	both are finitely generated abelian groups	$H_{p}(M_{1})\oplus H_{p}(M_{2})$
$n-1$	Both manifolds are compact, at least one of them is orientable	both are finitely generated abelian groups, at least one is free abelian	$H_{n-1}(M_{1})\oplus H_{n-1}(M_{2})$
$n-1$	other cases	?	?
$n$	Both are compact and orientable	both are $\mathbb {Z}$	$\mathbb {Z}$
$n$	other cases	?	?
Greater than $n$	none	both are zero groups	0

Euler characteristic

The Euler characteristics are related by the following formula when both $M_{1}$ and $M_{2}$ are compact connected manifolds:

$\chi (M_{1}\#M_{2})=\chi (M_{1})+\chi (M_{2})-\chi (S^{n})$

@@ Line 1: / Line 1: @@
-===Homology in low and high dimensions===
+{{effect of operation|connected sum|homology groups}}
-In all dimensions other than <math>n</math> and <math>n-1</math>, we have the following formula:
+==Statement==
-<math>\tilde{H}_i(M_1 \sharp M_2) = \tilde{H}_i(M_1) \oplus \tilde{H}_i(M_2)</math>
+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:
-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>.
+{| 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>
-===In the second highest dimension===
+|-
+| 0 || none || both isomorphic to <math>\mathbb{Z}</math> || <math>\mathbb{Z}</math> (because both are connected, so is their connected sum).
-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.
+|-
+| 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>
-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> || 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>
-<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>n-1</math> || other cases || ? || ?
-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> || Both are compact and orientable || both are <math>\mathbb{Z}</math> || <math>\mathbb{Z}</math>
-<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>
+|-
+| <math>n</math> || other cases || ? || ?
-is exact, and this yields, along with Mayer-Vietoris, that:
+|-
+| Greater than <math>n</math> || none || both are zero groups || 0
-<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===
@@ Line 33: / Line 27: @@
 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>