Twisted K-theory - What Is IT?

What Is It?

When H is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However when H is nontrivial this theory is no longer a ring. It has an addition, but it is no longer closed under multiplication.

However, the direct sum of the twisted K-theories of M with all possible twists is a ring. In particular, the product of an element of K-theory with twist H with an element of K-theory with twist H' is an element of K-theory twisted by H+H'. This element can be constructed directly from the above definition by using adjoints of Fredholm operators and construct a specific 2 x 2 matrix out of them (see the reference 1, where a more natural and general Z/2-graded version is also presented). In particular twisted K-theory is a module over classical K-theory.