combinatorial numbers appearing in the topology of manifolds

Question

This is a somewhat soft question. I would like to know examples when interesting numbers from combinatorics appear naturally or even unexpectedly in the study of manifolds and their topological invariants. A basic example which captures what I am looking for is the formula $b_{k}(T^{n}) = C(n,k)$ i.e. we see the most fundamental numbers from combinatorics appearing in the Betti numbers of the torus.
Note: It is already clear to me that combinatorics is an extremely useful tool for the study of manifolds, due triangulations, CW complexes etc. However, what I am looking for in this question is somehow in the opposite direction, i.e. interesting combinatorial numbers appearing in the topology of manifolds without any reference to a "combinatorial" structure.  I already have some more sophisticated examples in mind than the one I gave, but I would to prefer not to bias the discussion with these.

Connor Malin · Accepted Answer

The Bernoulli numbers come up every so often in algebraic topology. A rather bizarre instance is that the number of exotic spheres that are the boundary of manifold with trivial tangent bundle can be almost exactly calculated. If $2m>2$, then up to at most a single factor of 2, the number of such $(4m-1)$-spheres is $2^{2m-2}(2^{2m-1}-1)$ multiplied by the numerator of $4B_m /m$ written in simplest form, where $B_m$ is the mth Bernoulli number. This was calculated by Kervaire and Milnor.
All the instances of the Bernoulli numbers I know in topology can be traced back to Adams' work on the image of the J homomorphism from the homotopy groups of $O$ to the stable homotopy groups of spheres. He calculated the size of the image in terms of Bernoulli numbers. It is relevant to this example because exotic spheres are all stably framed and such stable framings are controlled by $O$.

combinatorial numbers appearing in the topology of manifolds

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