# Projective planes Joint work with Zhixu Su

The Hirzebruch -polynomial is one place where number theory very strongly interacts with high-dimensional topology [1]. Recall that the Hirzebruch signature theorem relates the signature of a smooth closed oriented manifold to . Unfortunately, the naïve method to compute coefficients of the Hirzebruch -polynomial is much too slow for applications; Zhixu Su and I have discovered a recursive method which is fast enough to compute many coefficients.

Solutions to some Diophantine equations related to these -polynomials give rise to manifolds having a truncated polynomial algebra as their rational cohomology ring; such manifolds may exist even when the corresponding truncated polynomial algebra over is not the cohomology ring of any space. For instance, there is a manifold having the rational cohomology that would be expected to have, if existed.

On the other hand, nonexistence results are also possible. A rational projective plane means a smooth manifold with with .

**Question.** For which is there a rational projective plane ?

There is a piecewise linear -homology manifold with as its cohomology ring, so this question can be viewed as a question about whether that -homology manifold can be “resolved” by a smooth manifold. Integrally, this is not always possible by the celebrated work of Adams [2], but in Zhixu Su’s thesis, it was shown that there *is* a rational projective plane in dimension 32. But it is not always possible, even rationally.

**Theorem (Fowler–Su).** There is no rational projective plane in dimension 64.

This boils down to some number theory. Since a rational projective plane has signature one, we would be seeking a solution to for integers and . Note that divides the numerator of , because divides —perhaps not so surprising considering 37’s status as the smallest irregular prime. So it is enough to show there is no solution to . Since ,

but neither nor is a quadratic residue modulo 37.

The same argument works to rule out a rational projective plane in dimension provided one can find a prime so that

and are quadratic nonresidues modulo ,

, but

.

To ensure is a quadratic nonresidue, it is enough that ; to ensure that is also a quadratic nonresidue, we further want .

The fact that divisors of are rarely (never?) tells us not to look for such primes among the divisors of the Mersenne factor. The above desiderata are satisfied by finding a prime so that

,

,

divides the numerator of ,

does not divide the numerator of .

The number theory becomes rather involved computationally. For example, the prime is and divides the numerator of but not , which rules out a rational projective plane in dimension ; similarly, the prime is , divides the numerator of but not , which rules out a projective plane in dimension . These calculations are possible due to tables of irregular primes produced by Joe P. Buhler and David Harvey [3].

Looking over these calculations, there are things that we can say in general.

**Theorem (Fowler–Su).** If is a rational projective plane, then for .

This result has some interesting consequences, such as the fact that there is a topological manifold which is not -homotopy equivalent to a smooth manifold. And even in dimensions for which there is a rational projective plane, a refined question can be explored.

**Question.** Suppose is a rational projective plane. How highly connected can be, integrally?

When , we have a particularly nice answer by doing -genus calculations: there does not exist a simply-connected closed Spin manifold which is a rational projective plane, so a 32-dimensional example cannot even be integrally 2-connected. Calculations involving the Steenrod algebra and Stiefel-Whitney classes may provide other methods for determining how highly connected a rational projective plane must be.

Finally, another interesting question is to study the asymptotic running time for algorithms computing the -polynomial.

**Question.** How quickly can the -polynomial be computed?

Such questions tie into Bernoulli number calculations, for which there are impressively fast analytic methods [4]. This would be a nice historical story, since the computation of Bernoulli numbers was the goal of computer program from 1843 written by Ada Lovelace [5].

[1] D.B. Zagier, Equivariant Pontrjagin classes and applications to orbit spaces. Applications of the -signature theorem to transformation groups, symmetric products and number theory, Springer-Verlag, Berlin-New York, 1972.

[2] J.F. Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39–41. https://doi.org/10.1090/S0002-9904-1962-10693-4.

[3] J.P. Buhler, D. Harvey, Irregular primes to 163 million, Math. Comp. 80 (2011) 2435–2444. https://doi.org/10.1090/S0025-5718-2011-02461-0.

[4] D. Harvey, A multimodular algorithm for computing Bernoulli numbers, Math. Comp. 79 (2010) 2361–2370. https://doi.org/10.1090/S0025-5718-2010-02367-1.

[5] A.K. Petrenko, O.L. Petrenko, The Babbage machine and the origins of programming, Istor.-Mat. Issled. (1979) 340–360, 389.