Ivancevic_Applied-Diff-Geom

524 Applied Differential Geometry: A Modern Introductionintegers, with −1 acting by multiplication. The circle is a model for K(Z, 1),and −1 acts by reflection. Using the action of Aut(Z) on K(Z, q) and thecocycle (4.28) we build an associated bundle H q → M with fiber K(Z, q).Equation (4.29) says that twisted cohomology classes are represented bysections of H q → M; the twisted cohomology group H q (M; Z) is the set ofhomotopy classes of sections of H q → M.Twistings may be defined for any generalized cohomology theory; ourinterest is in complex K−theory [Freed (2001); Freed et. al. (2003)]. Inhomotopy theory one regards K as a marriage of a ring and a space (moreprecisely, spectrum), and it makes sense to ask for the units in K, denotedGL 1 (K). In the previous paragraph we used the units in integralcohomology, the group Z mod 2. For complex K−theory there is a richergroup of unitsGL 1 (K) ∼ Zmod2 × CP ∞ × BSU. (4.30)In our problem the last factor does not enter and all the interest is in thefirst two, which we denote GL 1 (K) ′ . As a first approximation, view Kas the category of all finite dimensional Z mod 2–graded complex vectorspaces. Then CP ∞ is the subcategory of even complex lines, and it is agroup under tensor product. It acts on K by tensor product as well. Thenontrivial element of Z mod 2 in (4.30) acts on K by reversing the parityof the grading. This model is deficient since there is not an appropriatetopology. One may consider instead complexes of complex vector spaces,or spaces of operators as we do below. Of course, there are good topologicalmodels of CP ∞ , for example the space of all 1D subspaces of a fixed complexHilbert space H. For a manifold M the twistings of K−theory of interestare classified up to isomorphism byH 1 (M; GL 1 (K) ′ ) ∼ = H 1 (M; Zmod2) × H 3 (M; Z).In this paper we will not encounter twistings from the first factor and willfocus exclusively on the second. These twistings are represented by co–cycles g ij with values in the space of lines, in other words by complex linebundles L ij → U ij which satisfy a cocycle condition. This is the data oftengiven to define a gerbe. 99 Recall that a gerbe is a construct in homological algebra. It is defined as a stackover a topological space which is locally isomorphic to the Picard groupoid of that space.Recall that the Picard groupoid on an open set U is the category whose objects are linebundles on U and whose morphisms are isomorphisms. A stack refers to any categoryacting like a moduli space with a universal family (analogous to a classifying space)