Connected sum

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[edit] 1 Connected sum of smooth manifolds

Let M_0 and M_1 be oriented closed smooth connected n-manifolds. Their connected sum is an oriented closed smooth connected 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 preserves orientations and 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) with i_1((1-t)u) for u \in S^{n-1} and 0 < t < 1. The smooth structure on M_0 \sharp M_1 is obtain from the charts on M_0 - i_0(0) and M_1 - i_1(0). The orientation on M_0 \sharp M_1 is chosen to be the one compatible with the orientation of M_0 and 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-disc into a closed oriented smooth 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 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' and M_1 be oriented closed connected smooth manifold. Suppose that M_0 is h-cobordant to M_0', resp. bordant to M_0' then M_0 \sharp M_1 is h-cobordant, resp. bordant, to M_0' \sharp M_1.

[edit] 2 Connected sum of topological manifolds

Connected sum is a well-defined operation up to orientation preserving homeomorphism for oriented closed connected topological 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.

[edit] 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 and N be locally oriented manifolds such that there is a diffeomoprhism N \cong -N, then 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 be the total space of the non-trivial 3-sphere bundle over S^4 with Euler class zero and Pontrjagin class four times a preferred generator of 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)
  2. M \sharp \Sigma \cong M for any homotopy sphere \Sigma. (Recall that the group of homotopy 7-spheres, \Theta_7 is isomorphic to \Z/28.)

Proof. 1.) The manifold M is the boundary of the total space of the corresponding disc bundle W : = D^4 \tilde \times S^4 and hence M \sharp M is the boundary of W \natural W. Compact 3-connected 8-manifolds were classified in [Wall1962a, Section 2]. Since the intersection form of 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). Restricting f to the boundary gives the desired diffeomorphism.

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


[edit] 4 Properties

Let M be a closed connected n-manifold and let

\displaystyle M^\bullet : = M \setminus \textup{Int}(D^n)

denote the compact manifold obtained from M by deleting a small embedded open 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 denotes the one point union of topological spaces and \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 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) .

[edit] 5 References

[edit] 6 External links

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