Parametric connected sum

1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds $M$$\newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \DeclareMathOrd\GL{GL} % general linear group \DeclareMathOrd\SL{SL} % special linear group \DeclareMathOrd\SO{SO} % special orthogonal group \DeclareMathOrd\SU{SU} % special unitary group \DeclareMathOrd\Spin{Spin} % Spin group \DeclareMathOrd\RP{\Rr\mathrm P} % real projective space \DeclareMathOrd\CP{\Cc\mathrm P} % complex projective space \DeclareMathOrd\HP{\Hh\mathrm P} % quaternionic projective space \DeclareMathOrd\Top{\mathrm{Top}} % topological category \DeclareMathOrd\PL{\mathrm{PL}} % piecewise linear category \DeclareMathOrd\Cat{\mathrm{Cat}} % any category \DeclareMathOrd\KS{KS} % Kirby-Siebenmann class \DeclareMathOrd\Hud{Hud} % Hudson torusM$ and $N$$N$ equipped with codimension 0-embeddings $\phi: T \to M$$\phi: T \to M$ and $\psi : T \to N$$\psi : T \to N$ of a compact connected manifold $T$$T$. It generalises the usual connected sum operation but is more subtle since the isotopy classes of the embeddings $\phi$$\phi$ and $\psi$$\psi$ may be significantly more complicated than the isotopy classes of embeddings of n-discs need for connected sum: these last are determined by (local) orientations.

2 Connected sum

Let $M$$M$ be a compact connected n-manifold with base point $x \in \mathrm{int}$$x \in \mathrm{int}$. Recall that that a local orientation for $M$$M$ is a choice of orientation of $TM_m$$TM_m$, the tangent space to $M$$M$ at $m$$m$. We write $-M$$-M$ for $M$$M$ with the opposition orientation at $m$$m$. Of course, if $M$$M$ is orientable then a local orientation for $M$$M$ defines an orientation on $M$$M$.

If $M$$M$ and $N$$N$ are locally oriented n-manifolds then their connected sum is defined by

$\displaystyle M \sharp N = ((M - m) \cup (N - n))/ \simeq$

where $\simeq$$\simeq$ is defined using the local orientations to identify small balls about $k$$k$ and $n$$n$. The diffeomorphism type of $M \sharp N$$M \sharp N$ is well-defined: in fact $M \sharp N$$M \sharp N$ is the outcome of 0-surgery on $M \sqcup N$$M \sqcup N$. The essential point is [Hirsch] which states, for any $M$$M$ and any two compatibly oriented embeddings $f_0: D^n_1 \to M$$f_0: D^n_1 \to M$ and $\phi_1 : D^n \to M$$\phi_1 : D^n \to M$, that $\phi_0$$\phi_0$ is isotopic to $f_1$$f_1$.

If $M$$M$ and $N$$N$ are oriented manifolds the connected sum $M \sharp N$$M \sharp N$ is a well-defined up to diffeomorphism. Note that orientation matters! The canoical 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 2.1. 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)$.

3 Connected sum along k-spheres

We say above that to define connected sum for connected k-manifolds $M$$M$ and $N$$N$ it is sufficient to equip them with an isotopy class of embeddings of the k-disc. Moreover, the disjoint union $D^n \sqcup D^n$$D^n \sqcup D^n$ is the unique thickening of $S^0$$S^0$. This motivates the following

Defintion 3.1. An $S^k$$S^k$-oriented manifold is a pair $(M, \phi)$$(M, \phi)$ where $M$$M$ is a compact connected manifold and $\phi : S^k \times D^{n-k} \to \mathrm{int}(M)$$\phi : S^k \times D^{n-k} \to \mathrm{int}(M)$ is an embedding.

Defintion 3.2. Let $M = (M, \phi)$$M = (M, \phi)$ and $N = (N, \psi)$$N = (N, \psi)$ by $S^k$$S^k$-oriented manifolds. Define

$\displaystyle M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/simeq$

where $\simeq$$\simeq$ is defined via the embeddings $\phi$$\phi$ and $\psi$$\psi$.

Is is clear that we have the following

Observation 3.3. The diffeomorphism type of $M \sharp_k N$$M \sharp_k N$ depends only upon the the isotopy classes of the embeddings $\phi$$\phi$ and $\psi$$\psi$ (which of course includes the diffeomorphism types of $M$$M$ and $N$$N$).

3.1 Applications

The operation of $S^k$$S^k$-connected sum was used in [Ajala1984] and [Ajala1987] to describe the set of smooth structures on the product of spheres $\Pi_{i=1}^r S^{n_i}$$\Pi_{i=1}^r S^{n_i}$. It is also used in [Skopenkov] to define, for appropriate values of $p, q$$p, q$ and m groups stuctures on $E^m(S^p \times S^q)$$E^m(S^p \times S^q)$ the set of smooth isotopy classes of embeddings of $S^p \times S^q$$S^p \times S^q$ into $\Rr^m$$\Rr^m$. It also appears in [Sako1981].

4 Parametric connected sum along thickenings

Let $B$$B$ be a stable fibred vector bundle. A foundational theorem of modified surgery is

Theorem 4.1 Stable classification: [Kreck1985], [Kreck1999].

$\displaystyle NSt_{2n}(B) \cong \Omega_{2n}^B.$

In particular, $NSt_{2n}(B)$$NSt_{2n}(B)$ has the structure of an abelian group. The question of whether there is a geometric definition of this group structure is taken up in [Kreck1985, Chapter 2, pp 26-7] where it is shown how to use parametric connected sum along thickenings to define an addition of stable diffeomorphism classes of closed 2n-B-manifolds.