4-manifolds: 1-connected

Latest revision as of 15:58, 19 April 2011

[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 $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; 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_forms|intersection form]] is the main algebro-topological invariant of simply-connected 4-manifolds. Technical remark: When we mention the term ''4-manifold'' without the explicit mention of topological or smooth we shall mean the larger class of topological 4-manifolds. </wikitex> == Construction and examples, their intersection forms == <wikitex>; ... </wikitex> === First examples === <wikitex>; 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 $$ q_{\mathbb{CP}^2} = ( \ 1 \ ) , $$ $$ q_{\overline{\mathbb{CP}^2}} = ( \, -1 \ ) , $$ $$ q_{S^2 \times S^2} = \begin{pmatrix} \ 0 \ & \ 1 \ \ 1 & 0 \end{pmatrix} , $$ $$ q_{\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}} = \begin{pmatrix} \ 1 \ & \ 0 \ \ 0 & -1 \end{pmatrix} . $$ 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. </wikitex> === Hypersurfaces in CP³ === <wikitex>; For an integer $d \geq 1$ we define a subset $S_d$ of $\mathbb{CP}^3$ by the formula $$ S_d = \{ X_0^d + X_1^d + X_2^d + X_3^d = 0 | [X_0:X_1:X_2:X_3] \in \mathbb{CP}^3 \} . $$ 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_intersections|complete intersection]]. By the [[Wikipedia:Lefschetz_hyperplane_theorem|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]], which states that the signature of a 4-manifold is given by one third of the evaluation of the Pontryagin class on the fundamental cycle, 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 Poincaré 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 $$ c(T\mathbb{CP}^3) = (1+h)^4 . $$ We can now apply the Whitney sum formula for the total Chern class to the splitting $T\mathbb{CP}^3 |_{S_d} = TS_d \oplus \nu_{S_d}$, $$ c(T\mathbb{CP}^3 |_{S_d}) = c(TS_d) c(\nu_{S_d}) = c(TS_d) (1+dh) $$ which we can invert to obtain the formula $$ c_(TS_d) = (1+h)^4 (1-dh + d^2 h^2) , $$ and in particular $$ c_1(TS_d) = (4-d)h, \text{ \ and \ \ }c_2(TS_d) = (6-4d +d^2) h^2. $$ 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 $$ b_2(S_d) = (6-4d+d^2)d - 2 \text{ \ and \ \ } \text{sign}(S_d) = \frac{1}{3} (4-d^2)d $$ Furthermore $S_d$ is spin if and only if $d$ is even. This is because we have $$ w_2(TS_d) = d h \ (\text{mod} \ 2) , $$ and because the inclusion $S_d \hookrightarrow \mathbb{CP}^3$ yields an injective restriction map in second cohomology with $\Z/2$ coefficients because of the hypersection theorem. 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|intersection forms]] we know that the intersection form of $S_4$ is given by $$ q_{S_4} = -2 E_8 \oplus 3 H . $$ </wikitex> === Elliptic surfaces === <wikitex>; </wikitex> === Branched coverings === <wikitex>; </wikitex> === The E₈ manifold === <wikitex>; </wikitex> == Invariants == <wikitex>; ... </wikitex> == Topological classification == <wikitex>; ... </wikitex> == Non-existence results for smooth 4-manifolds == <wikitex>; ... </wikitex> == The Seiberg-Witten invariants == <wikitex>; ... </wikitex> == Failure of the h-cobordism theorem == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]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 Poincaré 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:

For the classification of topological 4-manifolds Freedman achieved to use surgery theory in order to establish his famous

[edit] 5 Non-existence results for smooth 4-manifolds

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:

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

• [Milnor&Husemoller1973] J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973. MR0506372 (58 #22129) Zbl 0292.10016