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The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the underlying setting of my questions is given by the local family index theorem with odd dimensional fibers. My questions are somehow related to the post "index of a family of Dirac operators in $K^1$".

The background of my questions is the following. Let $\pi:X\to B$ be a proper submersion whose fibers are odd dimensional oriented spin manifolds and $E\to X$ a complex vector bundle. After putting the necessary geometric data (metrics, horizontal distribution, various unitary connections), we have a spin Dirac operator $D$. To define the analytic index $\textrm{ind}(D)\in K^{-1}(B)$, we define an operator $D_\theta$ following (3.2) of page 81 of Atiyah-Patodi-Singer's paper "Spectral asymmetry and Riemannian geometry III". Then we get an index $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$. Since its restriction to $\{0\}\times B$ is trivial, it lies in $\ker(i^*:K^0(\mathbb{S}^1\times B)\to K^0(B))\cong K^{-1}(B)$, where $i:B\to\mathcal{S}^1\times B$ is the inclusion map, and $\textrm{ind}(D)\in K^{-1}(B)$ is defined to be $$\textrm{ind}(D)=\textrm{ind}(D_\theta)\in K^{-1}(B).$$ More precisely, the right-hand side of the above should be regarded as the corresponding element.

My questions are

  1. If, by some luck, the family of kernels $\ker(D_b)$ form a vector bundle (which is ungraded), does $\ker(D)\to B$ represent $\textrm{ind}(D)\in K^{-1}(B)$?

  2. If the answer to Q.1 is yes, what can we say about $\ker(D)\to B$ and $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$? Or more precisely, if $\ker(D)\to B$ exists, does $\ker(D_\theta)\to B$ exist? If so, how are they related?

  3. Let's suppose $\ker(D_b)$ do not form a bundle. It's well known that in the even-dimensional fiber case if $\ker(D_\theta)\to B$ does not exist one can perturb $D_\theta$ so that $\textrm{ind}(D_\theta)=[\ker(D_\theta')]-[\mathbb{C}^n]$, where $D_\theta'$ is the perturbed operator and $[\mathbb{C}^n]$ means it is represented by a trivial bundle of rank $n$ for some $n$. Is there any "perturbation method" in the odd dimensional fiber case? So I meant, is there any way to define $\textrm{ind}(D)\in K^{-1}(B)$ directly even though $\ker(D_b)$ do not form a bundle (this kind of implicitly assumes that the answer to Q.1 is yes) without going through the "suspension trick"?

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  • $\begingroup$ If $\ker (D)$ forms a vector bundle, then necessarily ind$(D)=0$. However, you could be lucky and have that the fibrewise spectrum intersects $0$ transversely. Then you get a kernel along a hypersurface in $B$. But this can only happen if the cup product of any two odd Chern classes of ind$(D)$ vanish. I suggest to have a glance at Eva Müller's preprint arXiv:1802.06568; thereis related work by Ebert, too. $\endgroup$ – Sebastian Goette Dec 26 '18 at 12:33
  • $\begingroup$ Thanks for the preprint by Eva M\"uller, the results are interesting and unexpected indeed. On the other hand, the proof of Theorem 4.1 in Ebert's paper is very precise and clear. $\endgroup$ – Ho Man Ho Dec 30 '18 at 9:44

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