Hirzebruch surfaces
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Contents |
1 Introduction
Hirzebruch surfaces were introduced by Hirzebruch (without that name) in his thesis [Hirzebruch1953a]. They are algebraic surfaces over the complex numbers. Here we consider them as smooth manifolds.
2 Construction and examples
Let for some be the tautological line bundle over . For define as the total space of the projective line bundle associated to , where is the tensor product of copies of with itself. 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
The dimension as a smooth manifold is .
Since the Hirzebruch surfaces are - as real manifold - -bundles over they are closed and - by the orientation coming from the complex structure - oriented -dimensional manifolds and the homotopy sequence of a fibration implies that they are simply connected.
The signature of is zero since the Hirzebruch surfaces are the boundary of the associated -bundle.
The homology is trivial except in degree and , where it is and in degree where it is isomorphic to with basis given by the homology class represented by a fibre and the homology class of the base considered as homology class of using the section of points at infinity.
The intersection form on is with respect to this basis given by the matrix
0 1
1 -n
The Euler characteristic is .
4 Classification/Characterization)
The intersection form implies that if is diffeomorphic to , then mod . On the other hand they are - as smooth manifolds - total spaces of the sphere bundle of a -dimensional vector bundle over . these bundles are classified by . Thus there are at most two diffeomorphism types of the fibre and so we conclude:
if and only if mod , where means diffeomorphic. By construction and by an easy consideration # .
For more information on Hirzebruch surfaces see [Hirzebruch1953a].
5 Further remarks
The Hirzebruch surfaces are the first stage of the so called Bott tower, which is inductively constructed staring from a point as the total space of a projective bundle associated to , where is a line bundle over a lower Bott tower. The classification of the Bott manifolds 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
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$ and $, where it is $\mathbb Z$ and in degree $ where it is isomorphic to $\mathbb Z^2$ with basis given by the homology class represented by a fibre and the homology class of the base considered as homology class of $H_n$ using the section of points at infinity. The intersection form on $H_2(H_n)$ is with respect to this basis given by the matrix 0 1 1 -n The Euler characteristic is $e(H_n) =4$. == Classification ==3 Invariants
The dimension as a smooth manifold is .
Since the Hirzebruch surfaces are - as real manifold - -bundles over they are closed and - by the orientation coming from the complex structure - oriented -dimensional manifolds and the homotopy sequence of a fibration implies that they are simply connected.
The signature of is zero since the Hirzebruch surfaces are the boundary of the associated -bundle.
The homology is trivial except in degree and , where it is and in degree where it is isomorphic to with basis given by the homology class represented by a fibre and the homology class of the base considered as homology class of using the section of points at infinity.
The intersection form on is with respect to this basis given by the matrix
0 1
1 -n
The Euler characteristic is .
4 Classification/Characterization)
The intersection form implies that if is diffeomorphic to , then mod . On the other hand they are - as smooth manifolds - total spaces of the sphere bundle of a -dimensional vector bundle over . these bundles are classified by . Thus there are at most two diffeomorphism types of the fibre and so we conclude:
if and only if mod , where means diffeomorphic. By construction and by an easy consideration # .
For more information on Hirzebruch surfaces see [Hirzebruch1953a].
5 Further remarks
The Hirzebruch surfaces are the first stage of the so called Bott tower, which is inductively constructed staring from a point as the total space of a projective bundle associated to , where is a line bundle over a lower Bott tower. The classification of the Bott manifolds 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
- [[Template: |[ ]]] {{ }}
Please modify these headings or choose other headings according to your needs.MediaWiki:Stub