Connected sum of manifolds: Difference between revisions

Latest revision as of 00:41, 29 July 2011

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. Further information: homotopy type of connected sum depends on choice of gluing map

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.

Fundamental group

Fundamental group of connected sum is free product of fundamental groups in dimension at least three: This fails in dimension two, because the circle $S^{1}$ has nontrivial fundamental group.
Connected sum of simply connected manifolds is simply connected

@@ Line 1: / Line 1: @@
 ==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#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. {{further|[[homotopy type of connected sum depends on choice of gluing map]]}}
 ==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.
@@ Line 9: / Line 13: @@
 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===
+==Fundamental group==
-In all dimensions other than <math>n</math> and <math>n-1</math>, we have the following formula:
-<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 [[deleted-point inclusion]] (inclusion of manifold minus a point into the manifold) induces isomorphism on all homologies uptil <math>n-2</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 # 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'' are non-orientable, however, then an exceptional situation occurs.
+* [[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]]
-===In the highest dimension===
+==Related notions==
-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.
+* [[Fiber sum]]
+* [[Symplectic sum]]
+* [[Knot sum]]