Connected sum of manifolds: Difference between revisions

Revision as of 18:35, 2 April 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.

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

Further information: Fundamental group of connected sum

The fundamental group of the connected sum of two manifolds is the free product of their fundamental groups. In particular, the fundamental group of a connected sum of simply connected manifolds is simply connected.

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 ==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.

Revision as of 18:35, 2 April 2011

Definition

Homology

Fundamental group

Related notions