Hirzebruch surfaces
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Revision as of 19:57, 17 September 2009
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Contents |
1 Introduction
Hirzebruch surfaces were introduced by Hirzebruch (without that name) in his thesis [Hirzebruch1951]. They are algebraic surfaces over the complex numbers. Here we consider them as smooth manifolds. The interest in them comes from Hirzebruch's result that as complex manifolds they are pairwise distinct whereas there are only two diffeomorphism types.
2 Construction and examples
Let for some be the tautological line bundle bundle over . Recall that if is a vector bundle over (real or complex) taking fibrewise the projective space yields a bundle , where if and only if there is a in the ground field such that , with fibres P(E_x), the associated projective bundle. If is a smooth vector bundle over a smooth manifold, then P(E) is a smooth manifold, or similarly if and are holomorphic, the total space is a complex manifold.
For define as , where is the tensor product of copies of with itself and denotes the trivial complex line bundle. For the bundle is by definition the trivial bundle. For we define as the corresponding construction with , the complex conjugated bundle, instead of . These are the . They come with a complex structure but we consider them as smooth manifolds.
3 Invariants
As smooth manifolds the Hirzebruch surfaces are -bundles over . Hence they are closed and, by the orientation coming from the complex structure, oriented 4-dimensional manifolds. Let denote a fibre -sphere and let denote the section of points at infinity.
- and .
- , in particular .
- has basis with and .
- With respect to the above basis the intersection form on is given by the following matrix:
- The Euler characteristic is given by .
- The signature vanishes: .
- The first Pontrjagin class of is zero: .
- The second Stiefel-Whitney class is given by and mod .
- is a spinable if and only if is even.
- For the complex manifold , the first Chern class , is given by and .
Explanation
- The computation of the homotopy groups of follows from the homotopy sequence of a fibration and the existence of the section .
- The homology groups of can be computed by decomposing where is the disc bundle associated to and using the Mayer-Vietoris sequence.
- The computation of the intersection form follows by inspecting the embedded -spheres which represent and their normal bundles: in particular we apply the fact that the self intersection number of a homology class represented by an embedded oriented submanifold is the evaluation of the Euler class of the normal bundle on the fundamental class the submanifold.
- The signature of is zero since the Hirzebruch surfaces are the boundary of the associated -bundle. One can also see this directly from the intersection form.
- The first Pontrjagin class vanishes as its evaluation on the fundamenatal class of is an oriented bordism invariant.
4 Classification
The intersection form implies that if is diffeomorphic to , then mod . On the other hand, considered as smooth manifolds, Hirzebruch surfaces are the total spaces of the 2-sphere bundle of a 3-dimensional vector bundle over . these bundles are classified by (note that is diffeomorphic to ). Thus there are at most two diffeomorphism types of Hirzebruch surfaces and so we conclude:
where means diffeomorphic. By construction and by an easy consideration # , where # is the connected sum and is with the opposite orientation.
For more information on Hirzebruch surfaces, in particular why they are pairwise distinct as complex manifolds, see [Hirzebruch1951].
5 Further remarks
The Hirzebruch surfaces are the second stage of the so called Bott towers, which are inductively constructed starting from a point as the total space of a projective bundle associated to , where is a line bundle over a lower Bott tower (for more details see [Choi&Masuda&Suh2008]). The classification of the Bott towers up to homeomorphism or diffeomorphism is an interesting open problem. In particular one can ask wether the integral cohomology ring determines the homeomorphism or diffeomorphism type.
6 References
- [Choi&Masuda&Suh2008] Template:Choi&Masuda&Suh2008
- [Hirzebruch1951] F. Hirzebruch, Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten, Math. Ann. 124 (1951), 77–86. MR0045384 (13,574e) Zbl 0043.30302