Chern Character Number Belongs to integer

Question

From Getzler's definition [1], we know the odd Chern character is the following map $$Ch:K^1(M)\to H^{odd}(M;\mathbb C),~g\mapsto \sum_{k\geqslant0}(-\frac1{2\pi\sqrt{-1}})^k\frac{k!}{(2k+1)!}Tr[(g^{-1}dg)^{2k+1}],$$ where $M$ is a closed manifold and $g$ is an element in the homotopy class of $[M,GL_N(\mathbb C)]$.

I have some question in my mind, but I am not sure whether it is solved or not.

Q:

• Is there an answer(conjecture)to say the odd Chern character number $<Ch(g),[M]>$ is an integer?

  In other words, for the even Chern character $ch:K(M\times S^1)\to H^{even}(M\times S^1,\mathbb C)$, is $<ch(\xi),[M\times S^1]>$ an integer, for any $\xi\in K(M\times S^1)$?

• Or there exists a counter-example?

PS: It is welcome to introduce the background of such a problem.

[1] Ezra Getzler, MR 1231957 The odd Chern character in cyclic homology and spectral flow, Topology 32 (1993), no. 3, 489--507.

Answer 1

Here is one example, I think it is right. $M:=\mathbb P^2\times S^1$, then $M\times S^1\cong \mathbb P^2\times \mathbb T^2$.

So, we can find the line bundle $\mathcal O(1)$ on $\mathbb P^2$ and $L$ on $\mathbb T^2$, with $c_1(L)=1$. Hence the pull-back vector bundle $pr^*_1\mathcal O(1)\oplus pr^*_2L$, where $pr_i$ denotes the canonical projection on the $i$th component of $M\times S^1$.

Thus, $<ch(pr^*_1\mathcal O(1)\oplus pr^*_2L),[M\times S^1]>=\frac12$.

I hope, it is right. Please let us check together.