Twistings for other cohomology theories

Question

Twistings in cohomology theories have a long history and have been used to great effect. The classical example is cohomology with local coefficients. Using this one can formulate Poincaré duality and the Thom isomorphism for unorientable manifolds. More recently, twisted K-theory has been linked to string theory (as B-fields) and the representation theory of loop groups (the Freed-Hopkins-Teleman theorem).

However, cohomology and K-theory are the only examples I know of where twistings are used regularly. What is known for twistings of other cohomology theories, e.g. the one associated to cobordism? In this case, one would expect a similarity to K-theory, from the Conner-Floyd theorem. Do these have geometric interpretations? What are the applications of such twistings?

Answer 1

Twisted forms exist for all multiplicative generalized cohomology theories. A nice paper which discusses a modern point of view for twists of homology, K-theory, and TMF is the following paper Twists of K-theory and TMF by Ando-Blumberg-Gepner.

If E is a generalized cohomology theory, represented by a spectrum also denoted E, then the E-cohomology of X coincides with the homotopy classes of maps

$$[ \Sigma^{-i} X, E] $$

i.e. the "E-valued functions on X". Morally, if E is a ring spectrum then we can talk about E-module spectra, and about E-lines: those E-module spectra which are equivalent to E, but not necessarily canonically so. With a sufficiently robust theory, we should be able to talk about bundles of spectra over X, and in particular E-line bundles over X. Then the usual E-cohomology of X can be thought of as the sections of the trivial E-line bundle over X. An E-twist is a possibly non-trivial E-line bundle over X, and twisted E-cohomology consists of the sections of this line bundle.

The tricky part is making this philosophical picture into something mathematically precise. The above paper is one way to do this.

In general the E-lines are classified by the space $BGL_1(E)$, which is the classifying space of the $A_\infty$-space $GL_1(E)$. This space is defined by the pull-back diagram

GL_1(E) --> &Omega&infinE
   |        |
   v        v
 &pi0(E)x --> &pi0(E)

From this you can read off the homotopy groups of $BGL_1(E)$ and you see that for $n \geq 2$ they agree with those of E, but are shifted in degree.

More generally, when E is a commutative ring spectrum, one can study the larger class of "E-lines" which are invertible E-modules. This requires a robust theory of spectra where you have a good notion of smash product over E. This leads to a larger classifying space of E-lines whose zeroth homotopy group is the Picard group Pic(E). Even more generally, you could consider bundles of general E-modules (not necessarily invertible) to be twists. There are probably applications of this, but I don't recall any off-hand.

As far as geometric descriptions go, you might be asking for too much. Even for K-theory it is only the simplest kinds of twists corresponding to the bottom few homotopy groups of $BGL_1(K)$ which appear to have a clear geometric description (e.g. in terms of super gerbes and clifford algebras). The higher twists of K-theory are more subtle and it is not a priori clear that they have a purely geometric description.