10 JOHN ROE The complex CX*(M) is equipped with the usual coboundary map of Alex- ander-Spanier cohomology [64], that is q+l dp(x0,.,. y xq+l)=z ^ ( - l ) V ( ^ o , - . . , £ , . . . ,xq+i). (2.8) (As usual, the 'hat' denotes omission of the specified term.) It is straightforward to verify that d does indeed map CXq to CXq+1 and that d2 = 0. Thus CX* is a complex, and we define the coarse cohomology HX*(M) to be the cohomology of this complex. (2.9) REMARKS: (i) What we have defined could more accurately be called the coarse co- homology with real coefficients. Replacing R by Z in the definition leads to the coarse cohomology with integer coefficients, and we could define it with more general coefficients if required. We will not develop these theories in detail in this paper, as the applications we have in mind are insensitive to torsion phenomena but it seems likely that they would repay further study. (ii) There are several ways in which the definition might be modified. We could replace the coarse complex by the subcomplex consisting of all continuous cocycles or even, if M is a smooth manifold, all smooth cocycles. We will see in Chapter 3 that these replacements will not alter the cohomology with real coefficients. (iii) Another possibility would be to replace the complex CX* by the sub- complex CX£ consisting of all totally antisymmetric cochains. We will see in Chapter 3 that the cohomology of CX£ is the same as that of CX*, the isomorphism being implemented by complete antisymmetriza- tion. This will be important for the construction in Chapter 4 of a map from coarse theory to cyclic theory. (2.10) DEFINITION: Lei C*(M) denote the complex whose q'th term is the space of all locally bounded Borel functions Mq+1 R, with the same boundary map d. It is well-known that this complex is acyclic in dimensions 0 if one chooses a fixed base-point o 6 M, then a contracting homotopy s: Cq —• Cq~l is given by stp(xi,... ,xq) - (p(o,xu... ,xq). The map s does not, however, preserve the subcomplex CX*(M) of C*(M), so CX*(M) may very well have cohomology. We will refer to C*(M) as the acyclic complex of M. (2.11) Th e character map : There is a natural map c from HX*(M) to the ordinary cohomology of M with compact supports, H*(M). This map may be defined in a number of equivalent ways, of which the simplest is to interpret
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