The Hirzebruch $L$-polynomial is one place where number theory very strongly interacts with high-dimensional topology [MR339202]. Recall that the Hirzebruch signature theorem relates the signature of a smooth closed oriented manifold $M^{4k}$ to $\sum_I L_I p_I(M)$. Unfortunately, the naïve method to compute coefficients $L_I$ of the Hirzebruch $L$-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 $L$-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 $\mathbb{Z}$ is not the cohomology ring of any space. For instance, there is a manifold having the rational cohomology that $\mathbb{O}P^4$ would be expected to have, if $\mathbb{O}P^4$ existed.

On the other hand, nonexistence results are also possible. A rational projective plane means a smooth manifold $M^{4n}$ with $H^\star(M;\mathbb{Q}) = \mathbb{Q}[x]/(x^3)$ with $|x| = 2n$.

Question. For which $n$ is there a rational projective plane $M^{4n}$?

There is a piecewise linear $\mathbb{Q}$-homology manifold with $\mathbb{Q}[x]/(x^3)$ as its cohomology ring, so this question can be viewed as a question about whether that $\mathbb{Q}$-homology manifold can be “resolved” by a smooth manifold. Integrally, this is not always possible by the celebrated work of Adams [MR133837], 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 $s_{8,8} x^2 \pm s_{16} y = \pm 1$ for integers $x$ and $y$. Note that $37$ divides the numerator of $s_{16}$, because $37$ divides $B_{32}$–-perhaps not so surprising considering 37's status as the smallest irregular prime. So it is enough to show there is no solution to $x^2 \not\equiv \pm 1/s_{8,8} \pmod{37}$. Since $s_{16} \equiv 0 \pmod{37}$, \begin{align*} s_{8,8} &\equiv \frac{{s_k}^2 - s_{2k}}{2} \pmod{37} \\ &\equiv \frac{{s_k}^2}{2} \pmod{37}, \end{align*} but neither $2$ nor $-2$ is a quadratic residue modulo 37.

The same argument works to rule out a rational projective plane in dimension $2^{k+3}$ provided one can find a prime $p$ so that- $2$ and $-2$ are quadratic nonresidues modulo $p$,
- $\nu_p(s_{2 \cdot 2^k}) > 0$, but
- $\nu_p(s_{2^k}) = 0$.

To ensure $2$ is a quadratic nonresidue, it is enough that $p \not\equiv \pm 1 \pmod 8$; to ensure that $-2$ is also a quadratic nonresidue, we further want $p \equiv 5 \pmod 8$.The fact that divisors of $2^{2^k - 1} - 1$ are rarely (never?) $5 \bmod 8$ tells us not to look for such primes among the divisors of the Mersenne factor. The above desiderata are satisfied by finding a prime $p$ so that

- $p \equiv 5 \pmod 8$,
- $p > 4 \cdot 2^k$,
- $p$ divides the numerator of $B_{4 \cdot 2^k}$,
- $p$ does not divide the numerator of $B_{2 \cdot 2^k}$.

The number theory becomes rather involved computationally. For example, the prime $p = 502261$ is $5 \bmod 8$ and divides the numerator of $B_{4 \cdot 2^{11}}$ but not $B_{2 \cdot 2^{11}}$, which rules out a rational projective plane in dimension $2^{14} = 4096$; similarly, the prime $p = 69399493$ is $5 \bmod 8$, divides the numerator of $B_{4 \cdot 2^{21}}$ but not $B_{2 \cdot 2^{21}}$, which rules out a projective plane in dimension $2^{24}$. These calculations are possible due to tables of irregular primes produced by Joe P. Buhler and David Harvey [MR2813369].

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

Theorem.(Fowler–Su) If $M^n$ is a rational projective plane, then $n = 2^a + 2^b$ for $a, b \in \mathbb{N}$.

This result has some interesting consequences, such as the fact that there is a topological manifold which is not $\mathbb{Q}$-homotopy equivalent to a smooth manifold. And even in dimensions $4n$ for which there is a rational projective plane, a refined question can be explored.

Question. Suppose $M^{4n}$ is a rational projective plane. How highly connected can $M^{4n}$ be, integrally?

When $4n = 32$, we have a particularly nice answer by doing $\hat{A}$-genus calculations: there does not exist a simply-connected closed Spin manifold $M^{32}$ 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 $L$-polynomial.

Question. How quickly can the $L$-polynomial be computed?

Such questions tie into Bernoulli number calculations, for which there are impressively fast analytic methods [MR2684369]. 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 [MR550674].