|the version used for publication as of 09:48, 1 April 2011 and the changes since publication.|
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
Recall that if is a complex vector bundle over , then taking the fibre-wise projective space yields the associated projective bundle:
The fibres of are complex projective spaces and if is a holomorphic vector bundle over a complex manifold then is a complex manifold. Moreover, if denotes the trivial complex line bundle then admits a canonical section
which takes each point of to the ``line at infinity`` in .
We identify with the unit complex numbers and recall that the -sphere, , admits the free action defined by the equation: . The quotient of this action is . For any integer define the complex line bundle whose total space is the following quotient of
Definition 2.1. For define the Hirzebruch surface . It is a complex manifold of complex dimension but we consider it as a smooth manifold of dimension .
The Hirzebruch surfaces are -bundles over . Hence they are closed and, by the orientation coming from the complex structure, oriented 4-dimensional manifolds.
We list some invariants of the manifolds with explanations below: let denote the fibre over .
- and .
- : in particular .
- for and for .
- 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: .
- For the complex manifold , the first Chern class , is given by and .
- The second Stiefel-Whitney class is given by mod and .
- is a spinable if and only if is even.
- 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 is the Euler class of [Milnor&Stasheff1974, Problem 11-C].
- 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 fundamental class of is an oriented bordism invariant [Milnor&Stasheff1974, Lemma 17.3].
- For the vaules of let denote the complex tangent bundle of a complex manifold then and . Moreover we have and and . Both of these equalities are justified by the fact that the first Chern class of a complex line bundle equals the Euler class of the underlying real -plane bundle and the fact that the Euler class can be evaluated by counting the oriented intersection points of a transverse section with the zero section. For the bundle we see that there is a well-defined transverse section with precisely positively oriented intersections with the zero section.
- For note that in general the Stiefel-Whitney classes are of a complex manifold are the mod reductions of the Chern classes [Milnor&Stasheff1974, Problem 14-B].
Theorem 4.1 [Hirzebruch1951]. For the smooth manifolds
where means diffeomorphic. Moreover as complex manifolds
where means complex diffeomorphic.
For the first statement we see that parity of the intersection form implies that if is diffeomorphic to , then mod . On the other hand the smooth Hirzebruch surfaces are the total spaces of the 2-sphere bundle of a 3-dimensional vector bundle over and these bundles are classified by (note that is diffeomorphic to ). Thus there are precisely two diffeomorphism types of Hirzebruch surfaces. 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 show that the smooth and both admit infinitely many inequivalent complex structures.
- The smooth manifolds and are examples of manifolds with isomorphic homotopy groups but distinct homotopy types.
- 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&Suh2010]). The classification of the Bott towers up to homeomorphism or diffeomorphism is an interesting open problem. In particular one can ask whether the integral cohomology ring determines the homeomorphism or diffeomorphism type as it does for Hirzebruch surfaces. For 3-stage Bott towers Choi, Masuda and Suh [Choi&Masuda&Suh2010, Theorem 1.4] prove that the cohomology ring determines the diffeomorphism type.
- The Hirzebruch surfaces give examples where the isotopy classes of certain diffeomorphims do not contain holomorphic maps (in this case because the diffeomorphisms do not preserve the first Chern class). For example, the connected sum of complex conjugation in both factors of is not isotopic to a holomorphic map.
- [Choi&Masuda&Suh2010] S. Choi, D. Masuda and D. Y. Suh, Topological classification of generalized Bott towers, Trans. Amer. Math. Soc. 362 (2010), no. 2, 1097–1112. MR2551516 (2011a:57050) Zbl 1195.57060
- [Hirzebruch1951] F. Hirzebruch, Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten, Math. Ann. 124 (1951), 77–86. MR0045384 (13,574e) Zbl 0043.30302
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504