# Connected sum

## 1 Connected sum of smooth manifolds

Let $M_0$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M_0$ and $M_1$$M_1$ be oriented closed smooth connected $n$$n$-manifolds. Their connected sum is an oriented closed smooth connected $n$$n$-manifold $\displaystyle M_1 \sharp M_2$

which is defined as follows (c.f. [Kervaire&Milnor1963, Section 2]). Choose smooth embeddings $\displaystyle i_0 \colon D^n \to M_0 \quad \text{and} \quad i_1 \colon D^n \to M_1$

where $i_1$$i_1$ preserves orientations and $i_2$$i_2$ reverses orientations. The connected sum is formed from the disjoint union $\displaystyle \bigl( M_0 - i_0(0) \bigr) \sqcup \bigl(M_1 - i_1(0) \bigr)$

by identifying $i_0(tu)$$i_0(tu)$ with $i_1((1-t)u)$$i_1((1-t)u)$ for $u \in S^{n-1}$$u \in S^{n-1}$ and $0 < t < 1$$0 < t < 1$. The smooth structure on $M_0 \sharp M_1$$M_0 \sharp M_1$ is obtain from the charts on $M_0 - i_0(0)$$M_0 - i_0(0)$ and $M_1 - i_1(0)$$M_1 - i_1(0)$. The orientation on $M_0 \sharp M_1$$M_0 \sharp M_1$ is chosen to be the one compatible with the orientation of $M_0$$M_0$ and $M_1$$M_1$.

A fundamental lemma of differential topology, [Palais1959, Theorem 5.5] and [Cerf1961] states that any two orientation preserving smooth embeddings of the $n$$n$-disc into a closed oriented smooth $n$$n$-manifold are isotopic. As a consequence we have the following lemma.

Lemma 1.1 [Kervaire&Milnor1963, Lemma 2.1] . The connected sum operation is well defined, associative and commutative up to orientation preserving diffeomoprhism. The sphere $S^n$$S^n$ serves as the identity element.

The connected sum operation also descends to give well-defined operations on larger equivalence classes of oriented manifolds.

Lemma 1.2 c.f. [Kervaire&Milnor1963, Lemma 2.2] . Let $M_0$$M_0$, $M_0'$$M_0'$ and $M_1$$M_1$ be oriented closed connected smooth manifold. Suppose that $M_0$$M_0$ is h-cobordant to $M_0'$$M_0'$, resp. bordant to $M_0'$$M_0'$ then $M_0 \sharp M_1$$M_0 \sharp M_1$ is h-cobordant, resp. bordant, to $M_0' \sharp M_1$$M_0' \sharp M_1$.

## 2 Connected sum of topological manifolds

Connected sum is a well-defined operation up to orientation preserving homeomorphism for oriented closed connected topological $n$$n$-manifolds. However, there is no analogue of the Palais/Cerf result and so the proof is more complicated. See the mathoverflow discussion cited below.

## 3 Examples

The orientation of the manifolds is important in general. The canonical example is $\displaystyle \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$

The manifolds are not even homotopy equivalent: the first has signature 2 the other signature 0. The following elementary lemma is often useful to remember.

Lemma 3.1. Let $M$$M$ and $N$$N$ be locally oriented manifolds such that there is a diffeomoprhism $N \cong -N$$N \cong -N$, then $M \sharp N \cong M \sharp (-N)$$M \sharp N \cong M \sharp (-N)$.

Connected sum decompositions of manifolds are far from being unique. For example, let $M = S^3 \tilde \times S^4$$M = S^3 \tilde \times S^4$ be the total space of the non-trivial 3-sphere bundle over $S^4$$S^4$ with Euler class zero and Pontrjagin class four times a preferred generator of $H^4(S^4; \Z) \cong \Z$$H^4(S^4; \Z) \cong \Z$.

Lemma 3.2 c.f.[Wilkens1974/75, Theorem 1] . There are diffeomorphisms

1. $M \sharp M \cong M \sharp (S^3 \times S^4)$$M \sharp M \cong M \sharp (S^3 \times S^4)$
2. $M \sharp \Sigma \cong M$$M \sharp \Sigma \cong M$ for any homotopy sphere $\Sigma$$\Sigma$. (Recall that the group of homotopy 7-spheres, $\Theta_7$$\Theta_7$ is isomorphic to $\Z/28$$\Z/28$.)

Proof. 1.) The manifold $M$$M$ is the boundary of the total space of the corresponding disc bundle $W : = D^4 \tilde \times S^4$$W : = D^4 \tilde \times S^4$ and hence $M \sharp M$$M \sharp M$ is the boundary of $W \natural W$$W \natural W$. Compact $3$$3$-connected $8$$8$-manifolds were classified in [Wall1962a, Section 2]. Since the intersection form of $W \natural W$$W \natural W$ is trivial, it is a simple consequence of Wall's classification that there is a diffeomorphism $f \colon W \natural W \cong W \natural (D^4 \times S^4)$$f \colon W \natural W \cong W \natural (D^4 \times S^4)$. Restricting $f$$f$ to the boundary gives the desired diffeomorphism.

2.) This is a special case of [Wilkens1974/75, Theorem 1]. $\square$$\square$

## 4 Properties

Let $M$$M$ be a closed connected $n$$n$-manifold and let $\displaystyle M^\bullet : = M \setminus \textup{Int}(D^n)$

denote the compact manifold obtained from $M$$M$ by deleting a small embedded open $n$$n$-disc. From the definition it is clear that $\displaystyle (M_0 \sharp M_1)^\bullet = (M_0^\bullet) \natural (M_1^\bullet) \simeq M_0^\bullet \vee M_1^\bullet.$

Here $\vee$$\vee$ denotes the one point union of topological spaces and $\simeq$$\simeq$ indicates that two spaces are homotopy equivalent. Applying Van Kampen's Theorem we immediately obtain the following lemma.

Lemma 4.1. Let the dimension $n$$n$ be three or greater. Then the fundamental group of a connected sum is the free product of the fundamental group of the components: $\displaystyle \pi_1(M_0 \sharp M_1) \cong \pi_1(M_0) \ast \pi_1(M_2) .$