For a finite CW complex $X$, the Chern character gives an isomorphism of finite-dimensional vector spaces: $$ ch : K^*(X)\otimes \mathbb{Q} \to H^*(X, \mathbb{Q}). $$ The vector space $V = H^*(X, \mathbb{Q})$ thus comes equipped with two natural maximal-rank lattices:
- $L_H = H^*(X, \mathbb{Z}) / T$ (where $T$ is the torsion),
- $L_K = ch(K^*(X) / T')$ (where $T'$ is the torsion).
What we can say about the relationship between $L_H$ and $L_K$?
Here is simple necessary relationship. Let $n=[\dim X/2]$; using denominators of size at most $n!$, the Chern character can be expressed as a linear combination of Chern classes. Chern classes are integral, so we must have: $$ n! \cdot L_K \subseteq L_H. $$
What other necessary conditions are known? Is there a known set of necessary and sufficient conditions? I.e., any $(V, L_H, L_K)$ satisfying them can be realised by some $X$.
For what it's worth, my motivation comes from spheres. Suppose we define a topological invariant $k$ to be the smallest natural number such that $k\cdot L_K \subseteq L_H$ (which presumably captures only a tiny bit of the relationship). Then the fact that $k=1$ for spheres allows a slick proof that the spheres admit no almost complex structure above dimension six. (In fact $L_K = L_H$ for spheres.)
Incidentally, the above ignores the $\mathbb{Z}/(2)$-grading. I haven't thought about this but I'd even be interested in the case when $X$ is even-dimensional with no odd-dimensional cohomology (e.g., a non-singular complex quadric).
