Connected sum
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1 Connected sum of smooth manifolds
Let and be oriented closed smooth connected -manifolds. Their connected sum is an oriented closed smooth connected -manifold
which is defined as follows (c.f. [Kervaire&Milnor1963, Section 2]). Choose smooth embeddings
where preserves orientations and reverses orientations. The connected sum is formed from the disjoint union
by identifying with for and . The smooth structure on is obtain from the charts on and . The orientation on is chosen to be the one compatible with the orientation of and .
A fundamental lemma of differential topology, [Palais1959, Theorem 5.5] [Cerf1961] states that any two orientation preserving smootgh embeddings of the -disc into a closed oriented smooth -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 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 , and be oriented closed connected smooth manifold. Suppose that is h-cobordant to , resp. bordant to then is h-cobordant, resp. bordant, to .
2 Connected sum of topological manifolds
Connected sum is a well-defined operation up to orientation preserving homeomorphism for oriented closed connected topological -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
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.
LetTex syntax errorand be locally oriented manifolds such that there is a diffeomoprhism , then .
Connected sum decompositions of manifolds are far from being unique. For example, let be the total space of the non-trivial 3-sphere bundle over with Euler class zero and Pontrjagin class four times a preferred generator of .
Lemma 3.2 c.f.[Wilkens1974/75, Theorem 1] . There are diffeomorphisms
- for any homotopy sphere . (Recall that the group of homotopy 7-spheres, is isomorphic to .)
Proof.
1.) The manifoldTex syntax erroris the boundary of the total space of the corresponding disc bundle and hence is the boundary of . Compact -connected -manifolds were classified
in [Wall1962a, Section 2]. Since the intersection form of is trivial, it is a simple consequence of Wall's classification that there is a diffeomorphism . Restricting to the boundary gives the desired diffeomorphism.
2.) This is a special case of [Wilkens1974/75, Theorem 1].
4 Properties
Tex syntax errorbe a closed connected -manifold and let
Tex syntax errorby deleting a small embedded open -disc. From the definition
it is clear that
Here denotes the one point union of topological spaces and indicates that two spaces are homotopy equivalent. Applying Van Kampen's Theorem we immediately obtain the following lemma.
Lemma 4.1. Let the dimension be three or greater. Then the fundamental group of a connected sum is the free product of the fundamental group of the components:
5 References
- [Cerf1961] J. Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227–380. MR0140120 (25 #3543) Zbl 0101.16001
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Palais1959] R. S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125–141. MR0116352 (22 #7140) Zbl 0092.30802
- [Wall1962a] C. T. C. Wall, Classification of -connected -manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR0145540 (26 #3071) Zbl 0218.57022
- [Wilkens1974/75] D. L. Wilkens, On the inertia groups of certain manifolds, J. London Math. Soc. (2) 9 (1974/75), 537–548. MR0383435 (52 #4316)
6 External links
< 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, \cite{Palais1959|Theorem 5.5} \cite{Cerf1961} states that any two orientation preserving smootgh embeddings of the $n$-disc into a closed oriented smooth $n$-manifold are isotopic. As a consequence we have the following lemma. {{beginthm|Lemma|\cite{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. {{endthm}} The connected sum operation also descends to give well-defined operations on larger equivalence classes of oriented manifolds. {{beginthm|Lemma|c.f. \cite{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-cobordism|h-cobordant]] to $M_0'$, resp. bordant to $M_0'$ then $M_0 \sharp M_1$ is h-cobordant, resp. bordant, to $M_0' \sharp M_1$. {{endthm}} == Connected sum of topological manifolds ==which is defined as follows (c.f. [Kervaire&Milnor1963, Section 2]). Choose smooth embeddings
where preserves orientations and reverses orientations. The connected sum is formed from the disjoint union
by identifying with for and . The smooth structure on is obtain from the charts on and . The orientation on is chosen to be the one compatible with the orientation of and .
A fundamental lemma of differential topology, [Palais1959, Theorem 5.5] [Cerf1961] states that any two orientation preserving smootgh embeddings of the -disc into a closed oriented smooth -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 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 , and be oriented closed connected smooth manifold. Suppose that is h-cobordant to , resp. bordant to then is h-cobordant, resp. bordant, to .
2 Connected sum of topological manifolds
Connected sum is a well-defined operation up to orientation preserving homeomorphism for oriented closed connected topological -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
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.
LetTex syntax errorand be locally oriented manifolds such that there is a diffeomoprhism , then .
Connected sum decompositions of manifolds are far from being unique. For example, let be the total space of the non-trivial 3-sphere bundle over with Euler class zero and Pontrjagin class four times a preferred generator of .
Lemma 3.2 c.f.[Wilkens1974/75, Theorem 1] . There are diffeomorphisms
- for any homotopy sphere . (Recall that the group of homotopy 7-spheres, is isomorphic to .)
Proof.
1.) The manifoldTex syntax erroris the boundary of the total space of the corresponding disc bundle and hence is the boundary of . Compact -connected -manifolds were classified
in [Wall1962a, Section 2]. Since the intersection form of is trivial, it is a simple consequence of Wall's classification that there is a diffeomorphism . Restricting to the boundary gives the desired diffeomorphism.
2.) This is a special case of [Wilkens1974/75, Theorem 1].
4 Properties
Tex syntax errorbe a closed connected -manifold and let
Tex syntax errorby deleting a small embedded open -disc. From the definition
it is clear that
Here denotes the one point union of topological spaces and indicates that two spaces are homotopy equivalent. Applying Van Kampen's Theorem we immediately obtain the following lemma.
Lemma 4.1. Let the dimension be three or greater. Then the fundamental group of a connected sum is the free product of the fundamental group of the components:
5 References
- [Cerf1961] J. Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227–380. MR0140120 (25 #3543) Zbl 0101.16001
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Palais1959] R. S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125–141. MR0116352 (22 #7140) Zbl 0092.30802
- [Wall1962a] C. T. C. Wall, Classification of -connected -manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR0145540 (26 #3071) Zbl 0218.57022
- [Wilkens1974/75] D. L. Wilkens, On the inertia groups of certain manifolds, J. London Math. Soc. (2) 9 (1974/75), 537–548. MR0383435 (52 #4316)