What is the growth rate of the number of unoriented cobordism classes?

Question

Let $\Omega_n^O$ denote the abelian group of cobordism classes of closed, unoriented manifolds of dimension $n$, and let $d(n) := \lvert\Omega_n^O\rvert$. What are the asymptotics of $d(n)$?

It's possible to express $\log_2 d(n)$ as a modified partition function: by work of Thom, $\Omega_n^O$ is an $\mathbb F_2$-vector space with a basis of the form $$\{x_{\ell_1,\dotsc,\ell_j}\mid \ell_1 + \dotsb + \ell_j = n, \ell_i \ne 2^r - 1\},\qquad\qquad (*)$$ so $\dim_{\mathbb F_2}\Omega_n^O = \log_2 d(n)$ is equal to the number of partitions of $n$ into numbers that are not one less than a power of 2. Hence if $P(n)$ denotes the usual partition function, $d(n) = O(2^{P(n)})$. Is this sharp? What about lower bounds?

Answer 1

If you denote by $P'(n)$ the number of partitions of $n$ with parts not of the form $2^k-1$, it holds that $$\log P'(n)\sim \pi \sqrt{\frac{2n}{3}}$$ just like for the usual partition function. This will be true whenever you allow the parts to come from a set of integers of asymptotic density 1. See for example Nathanson's paper: "Asymptotic density and the asymptotics of partition functions".