Connected sum of manifolds: Difference between revisions

Revision as of 14:17, 3 December 2007

Definition

Let $M_{1}$ and $M_{2}$ be connected manifolds. A connected sum of $M_{1}$ and $M_{2}$ , denoted $M_{1} ♯ M_{2}$ , is constructed as follows. Let $f_{i} : R^{n} \to U_{i}$ be homeomorphisms where $U_{i}$ are open subsets of $M_{i}$ . Let ${M_{i^{'}}}^{'}$ denote the complement in $M_{i}$ of the image of the open unit ball in $R^{n}$ , under $f_{i}$ . Then the connected sum is the quotient of $M_{1} ⊔ M_{2}$ under the identification of the boundary $S^{n - 1}$ s with each other, via the composite $f_{2} \circ f_{1}^{- 1}$ .

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

Further information: Homology of connected sum

The homology of the connected sum can be computed using the Mayer-Vietoris homology sequence for open sets obtained by enlarging the ${M_{i^{'}}}^{'}$ s slightly, and using the fact that ${M_{i^{'}}}^{'}$ is a strong deformation retract of $M_{i}$ minus a point.

The interesting phenomena occur at $n$ and $n - 1$ , because this is where the gluing is occurring.

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 ==Homology==
+{{further|[[Homology of connected sum]]}}
 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.
-===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>