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


[edit] 2.1 First examples

The first examples that come to one's mind are the 4-sphere S^4, the complex projective space \mathbb{CP}^2, the complex projective space with its opposite (non-complex) orientation \overline{\mathbb{CP}^2}, the product S^2 \times S^2, various connected sums of these, and in particular \mathbb{CP}^2 \# \overline{\mathbb{CP}^2}.

The intersection form of the 4-sphere is the "empty form" of rank 0. The intersection forms of the others are given by

\displaystyle  \begin{array}{ccc} q_{\mathbb{CP}^2}  & =  & ( \ 1 \ ) , \\  \\ q_{\overline{\mathbb{CP}^2}} & =  & ( \, -1 \ ) , \\ \\ q_{S^2 \times S^2} \ & = & \ \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0  \end{pmatrix} , \\ \\ q_{\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}} \ & = & \ \begin{pmatrix} \ 1 \ & \ 0 \ \\ 0 & -1  \end{pmatrix} . \end{array}

The manifolds S^2 \times S^2 and \mathbb{CP}^2 \# \overline{\mathbb{CP}^2} 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 CP3

For an integer d \geq 1 we define a subset S_d of \mathbb{CP}^3 by the formula

\displaystyle  S_d = \{ X_0^d + X_1^d + X_2^d + X_3^d = 0 | [X_0:X_1:X_2:X_3] \in \mathbb{CP}^3 \} .

It is easy to check that in each chart of \mathbb{CP}^3 the S_d is cut out transversally by the homogeneous polynomial of degree d. Therefore, X_d 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 S_d is simply connected. Its intersection form may be computed as follows: First one computes the Chern classes of S_d. Evaluating the second Chern class on the fundamental class [S_d] yields the Euler characteristic and therefore the rank of H^2(S_d;\mathbb{Z}). Likewise, by computing the Pontryagin class and using the Hirzebruch signature theorem, stating that for a closed oriented 4-manifold X one has

\displaystyle   \text{sign}(X) = \frac{1}{3} \langle p_1(TX), [X] \rangle \ ,

one computes the signature of S_d. Whether the intersection form is even or odd may be seen from the second Stiefel-Whitney class w_2(S_d) = c_1(S_d) \ (\text{mod} \ 2).

There are three facts that we need to use:

  • The normal bundle \nu_{S_d} of S_d in \mathbb{CP}^3 is given by H^d, where H is the line bundle dual to the hyperplane \mathbb{CP}^2 \subseteq \mathbb{CP}^3. Its first Chern class h generates the cohomology ring of \mathbb{CP}^3.
  • The hypersurface S_d is PoincarĂ© dual to the class d h, or equivalently \langle h^2,[S_d] \rangle = d.
  • The total Chern class of S_d is given by
\displaystyle  c(T\mathbb{CP}^3) = (1+h)^4 .

We can now apply the Whitney sum formula for the total Chern class to the splitting T\mathbb{CP}^3  |_{S_d} = TS_d \oplus \nu_{S_d},

\displaystyle  c(T\mathbb{CP}^3  |_{S_d})  = c(TS_d) c(\nu_{S_d}) = c(TS_d) (1+dh)

which we can invert to obtain the formula

\displaystyle  c(TS_d) = (1+h)^4 (1-dh + d^2 h^2) ,

and in particular

\displaystyle  c_1(TS_d) = (4-d)h, \; \text{ and } \;\; c_2(TS_d) = (6-4d +d^2) h^2.

We compute the Euler characteristic \chi(S_d) = \langle c_2(S_d), [S_d] \rangle = (6-4d+d^2)d by the above mentioned fact. The first Pontryagin class p_1(TS_d) = -c_2(TS_d \oplus \overline{TS_d}) = (4-d^2) h^2 yields the signature \text{sign}(S_d) = \frac{1}{3} (4-d^2)d. We summarise

\displaystyle  b_2(S_d) = (6-4d+d^2)d - 2  \; \text{ and  } \;\; \text{sign}(S_d) = \frac{1}{3} (4-d^2)d

Furthermore S_d is spin if and only if d is even. This is because we have

\displaystyle  w_2(TS_d) = d h \ (\text{mod} \ 2) ,

and because the inclusion S_d \hookrightarrow \mathbb{CP}^3 yields an injective restriction map in second cohomology with \mathbb{Z}/2 coefficients because of the hypersection theorem.

Summarising the above discussion, we have

\displaystyle  q_{S_d} = \frac{1}{3}(d^3-6d^2+11d-3)\ (\ 1 \ ) \oplus \frac{1}{3} (2d^3 - 6d^2 +7d -3) \ (\ -1 \ )

for d odd, and

\displaystyle  q_{S_d} = \frac{1}{24} d(d^2-4)\, (-E_8) \, \oplus  \frac{1}{3}(d^3-6d^2 +11d -3) \, H

A particularly interesting special case is that of d=4. The surface S_4 is a K3 surface. It is spin, has signature -16, and has b_2=22. By the classification results of indefinite intersection forms we know that the intersection form of S_4 is given by

\displaystyle  q_{S_4} = -2 E_8 \oplus 3 H .

Blowing up the surface S_4 yields the manifold S_4 \# \overline{\mathbb{CP}^2} which now has an odd intersection form given by

\displaystyle  q_{S_4 \# \overline{\mathbb{CP}^2}} = 3 \ (\ 1\ ) \oplus 20 \ (\ -1\ ),

the same form as that of the 4-manifold 3 \mathbb{CP}^2 \#  20 \overline{\mathbb{CP}^2}. 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 E8 manifold

[edit] 3 Invariants


[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:

Theorem 4.1 (Milnor, Whitehead). Two simply connected 4-manifolds are homotopy equivalent if and only if they have isomorphic intersection forms.

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 q over \mathbb{Z} there is, up to homeomorphism, a unique simply connected topological 4-manifold with intersection form q.
  • Given any odd unimodular symmetric bilinear form q over \mathbb{Z} there are, up to homeomorphism, precisely two simply connected topological 4-manifolds with intersection form q. 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 X with even intersection form and \text{sign}(X)/8 \ \equiv 1 \ (\text{mod} \ 2), as for instance the E_8 manifold X_{E_8}. By Rohlin's theorem these manifolds cannot admit a smooth structure. However, the manifold X_{E_8} \# X_{E_8} 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 X be a smooth closed 4-manifold with definite intersection form q_X. Then q_X is diagonal, i.e. it is the direct sum of the 1-dimensional forms ( \ 1 \ ) in the positive definite case, and of the forms ( \ -1 \ ) in the negative definite case.

By Donaldson's theorem, the topological 4-manifold X_{E_8} \# X_{E_8} does not admit a smooth structure either, neither does any of the manifolds X_{E_8} \# X_{E_8} \# \mathbb{CP}^2 \# \dots \# \mathbb{CP}^2 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 k E_8 \oplus k (\ 1 \ ) with different values of k, l \in \mathbb{N} are isomorphic.

[edit] 6 The Seiberg-Witten invariants


[edit] 7 Failure of the h-cobordism theorem


[edit] 8 Further discussion


[edit] 9 References

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