Hirzebruch surfaces

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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. 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 $L:= \{[x],y)\in \mathbb {CP}^1 \times \mathbb C^2 |y = \lambda x$$<!-- COMMENT: To achieve a unified layout, besides using the template below, please OBSERVE the following: Besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use the format {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}} - For references, use e.g. {{cite|Milnor1958b}} END OF COMMENT --> {{Authors|Matthias Kreck}} == Introduction == <wikitex>; Hirzebruch surfaces were introduced by Hirzebruch (without that name) in his thesis {{cite|Hirzebruch1953a}}. 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. </wikitex> == Construction and examples == <wikitex>; Let $L:= \{[x],y)\in \mathbb {CP}^1 \times \mathbb C^2 |y = \lambda x$ for some $\lambda \in \mathbb C\}$ be the tautological line bundle over $\mathbb {CP}^1$. Recall that if $E$ is a vector bundle over $X$ (real or complex) taking fibrewise the projective space yields a bundle $P(E):= (E-0)/\sim$, where $v\sim w$ if and only if there is a $\lambda $ in the ground field such that $\lambda v = w$, with fibres P(E_x), the associated projective bundle. If $E$ is a smooth vector bundle over a smooth manifold, then P(E) is a smooth manifold, or similarly if $E$ and $X$ are holomorphic, the total space is a complex manifold. For $n \ge 0$ define $H_n$ as $P(L^n \oplus \mathbb C)$, where $L^n$ is the tensor product of $n$ copies of $L$ with itself. For $n =0$ the bundle $L^0$ is by definition the trivial bundle. For $n <0$ we define $H_n$ as the corresponding construction with $\bar L$, the complex conjugated bundle, instead of $L$. These are the $Hirzebruch$ $surfaces$. They come with a complex structure but we consider them as smooth manifolds. </wikitex> == Invariants == <wikitex>; The dimension as a smooth manifold is $. Since the Hirzebruch surfaces are - as real manifold - $S^2$-bundles over $S^2$ 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 $H_n$ is zero since the Hirzebruch surfaces are the boundary of the associated $D^3$-bundle (or derive this directly from the intersection form below). The homology is trivial except in degree <script type="math/tex">L:= \{[x],y)\in \mathbb {CP}^1 \times \mathbb C^2 |y = \lambda x$ for some $\lambda \in \mathbb C\}$$\lambda \in \mathbb C\}$ be the tautological line bundle over $\mathbb {CP}^1$$\mathbb {CP}^1$. Recall that if $E$$E$ is a vector bundle over $X$$X$ (real or complex) taking fibrewise the projective space yields a bundle $P(E):= (E-0)/\sim$$P(E):= (E-0)/\sim$, where $v\sim w$$v\sim w$ if and only if there is a $\lambda$$\lambda$ in the ground field such that $\lambda v = w$$\lambda v = w$, with fibres P(E_x), the associated projective bundle. If $E$$E$ is a smooth vector bundle over a smooth manifold, then P(E) is a smooth manifold, or similarly if $E$$E$ and $X$$X$ are holomorphic, the total space is a complex manifold.

For $n \ge 0$$n \ge 0$ define $H_n$$H_n$ as $P(L^n \oplus \mathbb C)$$P(L^n \oplus \mathbb C)$, where $L^n$$L^n$ is the tensor product of $n$$n$ copies of $L$$L$ with itself. For $n =0$$n =0$ the bundle $L^0$$L^0$ is by definition the trivial bundle. For $n <0$$n <0$ we define $H_n$$H_n$ as the corresponding construction with $\bar L$$\bar L$, the complex conjugated bundle, instead of $L$$L$. These are the $Hirzebruch$$Hirzebruch$ $surfaces$$surfaces$. They come with a complex structure but we consider them as smooth manifolds.

3 Invariants

The dimension as a smooth manifold is $4$$4$.

Since the Hirzebruch surfaces are - as real manifold - $S^2$$S^2$-bundles over $S^2$$S^2$ they are closed and - by the orientation coming from the complex structure - oriented $4$$4$-dimensional manifolds and the homotopy sequence of a fibration implies that they are simply connected.

The signature of $H_n$$H_n$ is zero since the Hirzebruch surfaces are the boundary of the associated $D^3$$D^3$-bundle (or derive this directly from the intersection form below).

The homology is trivial except in degree $0$$0$ and $4$$4$, where it is $\mathbb Z$$\mathbb Z$ and in degree $2$$2$ where it is isomorphic to $\mathbb Z^2$$\mathbb Z^2$ with basis given by the homology class represented by a fibre and the homology class of the base considered as homology class of $H_n$$H_n$ using the section of points at infinity.

The intersection form on $H_2(H_n)$$H_2(H_n)$ is with respect to this basis given by the matrix

0 1

1 -n

Here we use that tht the self intersection number of a class represented by an embedded oriented submanifold is the evaluation of the Euler class of the normal bundle on the fundamental class.

The Euler characteristic is $e(H_n) =4$$e(H_n) =4$.

The first Pontrjagin class of $H_n$$H_n$ is zero (its evaluation on the fundamental class is a bordism invariant) and the second Stiefel-Whitney class $w_2(H_n)$$w_2(H_n)$ evaluates trivial on the first base element and by $n$$n$ mod $2$$2$ on the second. In particular $H_n$$H_n$ is a Spin-manifold if and only if $n$$n$ is even. If we consider them as complex manifolds then we have the first Chern class $c_1(H_n)$$c_1(H_n)$, which evaluates trivial on the first base element and by $-n$$-n$ on the second.

4 Classification

The intersection form implies that if $H_n$$H_n$ is diffeomorphic to $H_m$$H_m$, then $n = m$$n = m$ mod $2$$2$. On the other hand they are - as smooth manifolds - total spaces of the sphere bundle of a $3$$3$-dimensional vector bundle over $S^2$$S^2$. these bundles are classified by $\pi_1(SO(3)) \cong \mathbb Z/2$$\pi_1(SO(3)) \cong \mathbb Z/2$ (note that $SO(3)$$SO(3)$ is diffeomorphic to $\mathbb {RP}^3$$\mathbb {RP}^3$). Thus there are at most two diffeomorphism types of the fibre and so we conclude:

$\displaystyle H_n \cong H_m$ $$\displaystyle H_n \cong H_m$$

if and only if $n = m$$n = m$ mod $2$$2$, where $\cong$$\cong$ means diffeomorphic. By construction $H_0 = S^2 \times S^2$$H_0 = S^2 \times S^2$ and by an easy consideration $H_1 = \mathbb {CP}^2$$H_1 = \mathbb {CP}^2$ # $- \mathbb {CP}^2$$- \mathbb {CP}^2$, where # is the connected sum and $-M$$-M$ is $M$$M$ with the opposite orientation.

For more information on Hirzebruch surfaces, in particular why they are pairwise distinct as complex manifolds, see [Hirzebruch1953a].

5 Further remarks

The Hirzebruch surfaces are the first stage of the so called Bott tower, which is inductively constructed starting from a point as the total space of a projective bundle associated to $L \oplus \mathbb C$$L \oplus \mathbb C$, where $L$$L$ is a line bundle over a lower Bott tower (for more details see Masuda). 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

• [Hirzebruch1953a] F. Hirzebruch, On Steenrod's reduced powers, the index of inertia, and the Todd genus, Proc. Nat. Acad. Sci. U. S. A. 39 (1953), 951–956. MR0063670 (16,159f) Zbl 0051.14501

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This page has not been refereed. The information given here might be incomplete or provisional.

$ 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 Here we use that tht the self intersection number of a class represented by an embedded oriented submanifold is the evaluation of the Euler class of the normal bundle on the fundamental class. The Euler characteristic is $e(H_n) =4$. The first Pontrjagin class of $H_n$ is zero (its evaluation on the fundamental class is a bordism invariant) and the second Stiefel-Whitney class $w_2(H_n)$ evaluates trivial on the first base element and by $n$ mod $ on the second. In particular $H_n$ is a Spin-manifold if and only if $n$ is even. If we consider them as complex manifolds then we have the first Chern class $c_1(H_n)$, which evaluates trivial on the first base element and by $-n$ on the second. == Classification == ; The intersection form implies that if $H_n $ is diffeomorphic to $H_m$, then $n = m$ mod $. On the other hand they are - as smooth manifolds - total spaces of the sphere bundle of a $-dimensional vector bundle over $S^2$. these bundles are classified by $\pi_1(SO(3)) \cong \mathbb Z/2$ (note that $SO(3)$ is diffeomorphic to $\mathbb {RP}^3$). Thus there are at most two diffeomorphism types of the fibre and so we conclude: $$ H_n \cong H_m$$ if and only if $n = m$ mod $, where $\cong$ means diffeomorphic. By construction $H_0 = S^2 \times S^2$ and by an easy consideration $H_1 = \mathbb {CP}^2$ # $- \mathbb {CP}^2$, where # is the connected sum and $-M$ is $M$ with the opposite orientation. For more information on Hirzebruch surfaces, in particular why they are pairwise distinct as complex manifolds, see {{cite|Hirzebruch1953a}}. == Further remarks == ; The Hirzebruch surfaces are the first stage of the so called Bott tower, which is inductively constructed starting from a point as the total space of a projective bundle associated to $ L \oplus \mathbb C$, where $L$ is a line bundle over a lower Bott tower (for more details see Masuda). 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. == References == {{#RefList:}} Please modify these headings or choose other headings according to your needs. [[Category:Manifolds]] {{Stub}}L:= \{[x],y)\in \mathbb {CP}^1 \times \mathbb C^2 |y = \lambda x for some $\lambda \in \mathbb C\}$$\lambda \in \mathbb C\}$ be the tautological line bundle over $\mathbb {CP}^1$$\mathbb {CP}^1$. Recall that if $E$$E$ is a vector bundle over $X$$X$ (real or complex) taking fibrewise the projective space yields a bundle $P(E):= (E-0)/\sim$$P(E):= (E-0)/\sim$, where $v\sim w$$v\sim w$ if and only if there is a $\lambda$$\lambda$ in the ground field such that $\lambda v = w$$\lambda v = w$, with fibres P(E_x), the associated projective bundle. If $E$$E$ is a smooth vector bundle over a smooth manifold, then P(E) is a smooth manifold, or similarly if $E$$E$ and $X$$X$ are holomorphic, the total space is a complex manifold.

For $n \ge 0$$n \ge 0$ define $H_n$$H_n$ as $P(L^n \oplus \mathbb C)$$P(L^n \oplus \mathbb C)$, where $L^n$$L^n$ is the tensor product of $n$$n$ copies of $L$$L$ with itself. For $n =0$$n =0$ the bundle $L^0$$L^0$ is by definition the trivial bundle. For $n <0$$n <0$ we define $H_n$$H_n$ as the corresponding construction with $\bar L$$\bar L$, the complex conjugated bundle, instead of $L$$L$. These are the $Hirzebruch$$Hirzebruch$ $surfaces$$surfaces$. They come with a complex structure but we consider them as smooth manifolds.

3 Invariants

The dimension as a smooth manifold is $4$$4$.

Since the Hirzebruch surfaces are - as real manifold - $S^2$$S^2$-bundles over $S^2$$S^2$ they are closed and - by the orientation coming from the complex structure - oriented $4$$4$-dimensional manifolds and the homotopy sequence of a fibration implies that they are simply connected.

The signature of $H_n$$H_n$ is zero since the Hirzebruch surfaces are the boundary of the associated $D^3$$D^3$-bundle (or derive this directly from the intersection form below).

The homology is trivial except in degree $0$$0$ and $4$$4$, where it is $\mathbb Z$$\mathbb Z$ and in degree $2$$2$ where it is isomorphic to $\mathbb Z^2$$\mathbb Z^2$ with basis given by the homology class represented by a fibre and the homology class of the base considered as homology class of $H_n$$H_n$ using the section of points at infinity.

The intersection form on $H_2(H_n)$$H_2(H_n)$ is with respect to this basis given by the matrix

0 1

1 -n

Here we use that tht the self intersection number of a class represented by an embedded oriented submanifold is the evaluation of the Euler class of the normal bundle on the fundamental class.

The Euler characteristic is $e(H_n) =4$$e(H_n) =4$.

The first Pontrjagin class of $H_n$$H_n$ is zero (its evaluation on the fundamental class is a bordism invariant) and the second Stiefel-Whitney class $w_2(H_n)$$w_2(H_n)$ evaluates trivial on the first base element and by $n$$n$ mod $2$$2$ on the second. In particular $H_n$$H_n$ is a Spin-manifold if and only if $n$$n$ is even. If we consider them as complex manifolds then we have the first Chern class $c_1(H_n)$$c_1(H_n)$, which evaluates trivial on the first base element and by $-n$$-n$ on the second.

4 Classification

The intersection form implies that if $H_n$$H_n$ is diffeomorphic to $H_m$$H_m$, then $n = m$$n = m$ mod $2$$2$. On the other hand they are - as smooth manifolds - total spaces of the sphere bundle of a $3$$3$-dimensional vector bundle over $S^2$$S^2$. these bundles are classified by $\pi_1(SO(3)) \cong \mathbb Z/2$$\pi_1(SO(3)) \cong \mathbb Z/2$ (note that $SO(3)$$SO(3)$ is diffeomorphic to $\mathbb {RP}^3$$\mathbb {RP}^3$). Thus there are at most two diffeomorphism types of the fibre and so we conclude:

$\displaystyle H_n \cong H_m$ $$\displaystyle H_n \cong H_m$$

if and only if $n = m$$n = m$ mod $2$$2$, where $\cong$$\cong$ means diffeomorphic. By construction $H_0 = S^2 \times S^2$$H_0 = S^2 \times S^2$ and by an easy consideration $H_1 = \mathbb {CP}^2$$H_1 = \mathbb {CP}^2$ # $- \mathbb {CP}^2$$- \mathbb {CP}^2$, where # is the connected sum and $-M$$-M$ is $M$$M$ with the opposite orientation.

For more information on Hirzebruch surfaces, in particular why they are pairwise distinct as complex manifolds, see [Hirzebruch1953a].

5 Further remarks

The Hirzebruch surfaces are the first stage of the so called Bott tower, which is inductively constructed starting from a point as the total space of a projective bundle associated to $L \oplus \mathbb C$$L \oplus \mathbb C$, where $L$$L$ is a line bundle over a lower Bott tower (for more details see Masuda). 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

• [Hirzebruch1953a] F. Hirzebruch, On Steenrod's reduced powers, the index of inertia, and the Todd genus, Proc. Nat. Acad. Sci. U. S. A. 39 (1953), 951–956. MR0063670 (16,159f) Zbl 0051.14501

Please modify these headings or choose other headings according to your needs.

This page has not been refereed. The information given here might be incomplete or provisional.