By Michael Spivak
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Extra resources for A Comprehensive Introduction to Differential Geometry, Vol. 5, Third Edition
4-term of our spectral sequence is just the part J + of J of positive degree: E4 =* J + . 9) Since there are no more terms of odd degree, all remaining differentials are zero, and so J+^Eoo^ gr HV(Y), where Hv is the universal twisted cohomology. To get characteristic twisted classes for twisted if-theory we therefore have to choose liftings of J + back into the filtered ring HV(Y). For example, for the first class x\ = s\ £ J2 the Chern character, as constructed in Section 7, provides a natural lift.
Segal 26 twisted cohomology of X. Although the latter question might seem more logical, we shall begin with the former, which is easier and more tractable. 8 below. For any generalized cohomology theory characteristic classes correspond to cohomology classes of the classifying space. For twisted if-theory the appropriate classifying space is the universal fibration studied in Section 3, with fibre Fred(H) and base the Eilenberg-MacLane space K(Z, 3). The cohomology classes of the total space define characteristic classes for our twisted if-theory.
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