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 \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. {{further|[[homotopy type of connected sum depends on choice of gluing map]]}}
 ==Homology==
@@ Line 12: / Line 12: @@
 The interesting phenomena occur at <math>n</math> and <math>n-1</math>, 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]] <math>S^1</math> has nontrivial fundamental group.
+* [[Connected sum of simply connected manifolds is simply connected]]
+==Related notions==
+* [[Fiber sum]]
+* [[Symplectic sum]]
+* [[Knot sum]]