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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).

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    $\begingroup$ Huh -- in algebraic geometry when there are two distinct lattices living in a vector space then you can take the determinant of both lattices and get an element of the ground field (their ratio), defined up to sign, which is a fundamental invariant of the object being studied. In algebraic geometry a conjecture of Deligne relates this number fo a special value of the L-function of the variety. So here this construction can be viewed as giving a fundamental positive rational number which is an invariant of the CW complex. What is this rational number? Are there other ways to compute it? $\endgroup$ Commented May 5, 2023 at 10:09

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I remember, dimly, work of J.Frank Adams,published in "Topology" (journal), in the early 1960's, on this kind of problem, Bruno Harris

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  • $\begingroup$ Thank you for the lead! I will chase it down at the weekend. $\endgroup$ Commented Oct 24, 2018 at 20:30

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