Suppose I have a family of Dirac operators over a compact base space B. From the paper of Atiyah and Singer about skew adjoint Fredholm operators we know that it has an index in $K^1(B)$.


Suppose furthermore I know "a lot" about these Dirac operators (like their spectrum, eigenspaces etc.) and $B$ is a simple space like e.g. a torus where I know everything about $K^1$ and the cohomology.


What methods are there to give an explicit description of the index in $K^1(B)$ or its image in the odd-dimensional cohomology?


Any suggestions or references welcome.



index of a family of Dirac operators in $K^1$
Whether the following is useful might depend on your concrete example. Because you are mentioning $K^1 $ instead of $K^0$, I assume that your Dirac operator is ungraded (if it is graded, the index should be in $K^0$). The graded case is the ordinary Atiyah-Singer family index theorem. There is a version for the ungraded case, with index in $K^1$, which is expressed by the same formula, except that the symbol class is in $K^1 $ and its definition is slightly more complicated than in the $K^0$-case. This is explained in Atiyah-Patodi-Singer, "Spectral asymmetry and Riemann geometry III" (you do not have to read parts I and II, but only one section of part III, nevertheless, these are beautiful papers). 
There is a cohomological formula that can be derived from the K-theory formula, much in the same way as in the ordinary $K^0$-case.
In other words, the family index in $K^1$ is as computable as the ordinary one. By the way: if you take a graded operator, you get - forget the grading - an ungraded operator. Its $K^1$-index is always zero.


If your base space is a circle and if you really know the spectrum of the operator at any point of $B$, there is a nice spectral-theoretic way for computing the index, by the so-called "spectral flow". The spectral flow of a map $f: S^1 \to Fred\_{sa}$ is the geometric intersection number of $f$ with the sub"manifold" of noninvertible operators in the space of selfadjoint Fredholms. Sounds complicated, but what you have to do is to count how often an eigenvalue crosses zero. A nice explanation of the spectral flow is in Booss-Woichechowski's book "Boundary value problems for Dirac-operators". 
If $B$ is a $2$-torus, then $K^1 (B)$ should be $Z^2$, the factor being detected on the two circles, and this reduces the problem to the case $B=S^1$.


A last comment: if some miracle happens and the dimension of the kernel is constant, then your computation is VERY easy. The family index in $K^1 (B)$ will be zero in that case.
I wrote down a simple proof of this fact in <http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.4719v3.pdf>, Theorem 4.2.1.


