Let $\mathbb{F}_n:=\mathbb{P}(\mathcal{O}(-n)\oplus\mathcal{O}(0))$ be the $n$th Hirzebruch surface, where $\mathcal{O}(k)$ is the canonical line bundle on $\mathbb{P}^1_\mathbb C$, for any $k\in\mathbb{Z}$. I'm trying to compute the homology groups of the surfaces $\mathbb F_n$. The unique reference I've found is this, but, in truth, it is unclear to me how to proceed. Can someone help me?
$\begingroup$
$\endgroup$
10
-
4$\begingroup$ The Betti numbers are 1, 0, 2, 0, 1 for all n, and there is no torsion. Probably the reference you have contains this. Are you asking you do you actually check this? $\endgroup$– Donu ArapuraCommented Sep 21, 2018 at 22:40
-
6$\begingroup$ @VincenzoZaccaro The Hirzebruch surfaces are $S^2$ bundles over $S^2$, and the spectral sequence for a fiber bundle will immediately give you the above homology groups. $\endgroup$– Aleksandar MilivojevićCommented Sep 21, 2018 at 22:49
-
2$\begingroup$ @VincenzoZaccaro A quick, practical introduction is given in Griffiths-Morgan “Rational Homotopy Theory and Differential Forms”; a much more in-depth reference is McCleary’s “A User’s Guide to Spectral Sequences”. $\endgroup$– Aleksandar MilivojevićCommented Sep 21, 2018 at 23:01
-
3$\begingroup$ You could also use Hurewicz to say $H_1 = 0$ and $H_2 = \mathbb{Z}^2$. The rest follows from this. (Here I'm using the fact that the $S^2$ bundle has a section.) $\endgroup$– Michael AlbaneseCommented Sep 21, 2018 at 23:08
-
3$\begingroup$ You could also use the well-known fact that the Hirzebruch surfaces with $n$ even are diffeomorphic to $S^2 \times S^2$ and the ones with $n$ odd are diffeomorphic to $\mathbb{C}P^2 \#\bar{\mathbb{C}}P^2$ (the blowup of $\mathbb{C}P^2$). This tells you the intersection form as well as the homology groups. $\endgroup$– Danny RubermanCommented Sep 22, 2018 at 11:39
|
Show 5 more comments
1 Answer
$\begingroup$
$\endgroup$
1
Various approach are suggested in the comments. Let me give some hints for an easy method. The nice thing about Hirzebruch, or rational ruled, surfaces is they are trivial over $V= \mathbb{P}^1-\{\infty\}$. So that the preimage $U$ is the product $\mathbb{P}^1\times V$. Let $Z=\mathbb{F}_n-U$. This is just a projective line. Now use the sequence $$\ldots H_c^i(U)\to H^i(\mathbb{F}_n)\to H^i(Z)\to H^{i+1}_c(U)\ldots$$ where the group on the left is compactly supported cohomology. Now you should have enough information to hopefully finish the computation, which would give $H^2(\mathbb{F_n})=\mathbb{Z}^2$ for example.
answered Sep 22, 2018 at 12:04