4-manifolds: 1-connected
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[edit] 1 Introduction
Any finitely presentable group may occur as the fundamental group of a smooth closed 4-manifold. On the other hand, the class of simply connected (topological or smooth) 4-manifolds still appears to be quite rich, so it appears reasonable to consider the classification of simply connected 4-manifolds in particular.
It appears that the intersection form is the main algebro-topological invariant of simply-connected 4-manifolds.
[edit] 2 Construction and examples, their intersection forms
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[edit] 2.1 First examples
The first examples that come to one's mind are the 4-sphere , the complex projective space , the complex projective space with its opposite (non-complex) orientation , the product , various connected sums of these, and in particular .
The intersection form of the 4-sphere is the "empty form" of rank 0. The intersection forms of the others are given by
The manifolds and both have indefinite intersection forms of same rank and signature, but of different type. Therefore they are not homotopy-equvialent.
[edit] 2.2 Hypersurfaces in CP^{3}
For an integer we define a subset of by the formula
It is easy to check that in each chart of the is cut out transversally by the homogeneous polynomial of degree . Therefore, is a submanifold, in fact, an algebraic hypersurface. This is a special case of a complete intersection.
By the Lefschetz hyperplane section theorem the hypersurface is simply connected. Its intersection form may be computed as follows: First one computes the Chern classes of . Evaluating the second Chern class on the fundamental class yields the Euler characteristic and therefore the rank of . Likewise, by computing the Pontryagin class and using the Hirzebruch signature theorem, stating that for a closed oriented 4-manifold one has
one computes the signature of . Whether the intersection form is even or odd may be seen from the second Stiefel-Whitney class .
There are three facts that we need to use:
- The normal bundle of in is given by , where is the line bundle dual to the hyperplane . Its first Chern class generates the cohomology ring of .
- The hypersurface is PoincarĂ© dual to the class , or equivalently .
- The total Chern class of is given by
We can now apply the Whitney sum formula for the total Chern class to the splitting ,
which we can invert to obtain the formula
and in particular
We compute the Euler characteristic by the above mentioned fact. The first Pontryagin class yields the signature . We summarise
Furthermore is spin if and only if is even. This is because we have
and because the inclusion yields an injective restriction map in second cohomology with coefficients because of the hypersection theorem.
Summarising the above discussion, we have
for odd, and
A particularly interesting special case is that of . The surface is a K3 surface. It is spin, has signature , and has . By the classification results of indefinite intersection forms we know that the intersection form of is given by
Blowing up the surface yields the manifold which now has an odd intersection form given by
the same form as that of the 4-manifold . Below we shall see that these two 4-manifolds are homeomorphic by Freedman's classification results, but not diffeomorphic because they have different Seiberg-Witten invariants.
[edit] 2.3 Elliptic surfaces
[edit] 2.4 Branched coverings
[edit] 2.5 The E_{8} manifold
[edit] 3 Invariants
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[edit] 4 Topological classification
By early work of Milnor and Whitehead the following theorem was known since 1958, and is based on a generalised Pontryagin-Thom construction:
For the classification of topological 4-manifolds Freedman achieved to use surgery theory in order to establish his famous
Theorem 4.2 (Freedman).
- Two simply-connected closed topological 4-manifolds are homeomorphic if and only if they have isomorphic intersection forms and the same Kirby-Siebenmann invariant.
- Given any even unimodular symmetric bilinear form over there is, up to homeomorphism, a unique simply connected topological 4-manifold with intersection form .
- Given any odd unimodular symmetric bilinear form over there are, up to homeomorphism, precisely two simply connected topological 4-manifolds with intersection form . One of them has non-trivial Kirby-Siebenmann invariant and therefore cannot be given a smooth structure.
[edit] 5 Non-existence results for smooth 4-manifolds
Theorem 5.1 (Rohlin). A smooth closed 4-manifold that is spin, and therefore has even intersection form, has its signature divisible by 16.
By Freedman's theorem we know that there are closed simply connected topological 4-manifolds with even intersection form and , as for instance the manifold . By Rohlin's theorem these manifolds cannot admit a smooth structure. However, the manifold might so because its signature is equal to 16.
Using methods from gauge theory Donaldson was able to prove his famous theorem on the intersection form of smooth definite 4-manifolds:
Theorem 5.2 (Donaldson). Let be a smooth closed 4-manifold with definite intersection form . Then is diagonal, i.e. it is the direct sum of the 1-dimensional forms in the positive definite case, and of the forms in the negative definite case.
By Donaldson's theorem, the topological 4-manifold does not admit a smooth structure either, neither does any of the manifolds with odd intersection form. In fact, it is an algebraic theorem due to Eichler [Milnor&Husemoller1973] that there is a unique decomposition theorem for definite forms, and so no two of the forms with different values of are isomorphic.
[edit] 6 The Seiberg-Witten invariants
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[edit] 7 Failure of the h-cobordism theorem
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[edit] 8 Further discussion
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[edit] 9 References
- [Milnor&Husemoller1973] J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973. MR0506372 (58 #22129) Zbl 0292.10016