VI - Sheaves

Summary

In this chapter we extend the treatment of coefficients in Chapter III to cover sheaves of abelian groups. We work always with pl cobordism but everything that we say can be extended to an arbitrary theory under the conditions of IV 6. 4. §§4 and 5 in fact extend unconditionally. The general definition of sheaves of coefficients does not have all the best properties one would hope for and we will explain where the difficulties lie at the start of §4.

In §1 we recall the basic properties of stacks and sheaves and in §2 we define the theory of mock bundles with coefficients in a stack. The definition is functorial on the category of all stacks of abelian groups. The main theorem asserts that, if the stack is ‘nice’, then there is a spectral sequence expressing the relation between simplicial cohomology and cobordism with coefficients in the stack. In §3 cobordism with coefficients in a sheaf is defined by means of a simplicial analogue of the Čech procedure. In §4 we discuss an extension of the methods used in the previous sections and give an example of ‘Poincaré duality’ between bordism and cobordism with coefficients in the sheaf of local homology of a Z_n-manifold. Finally in §5 we extend the methods further and give examples which suggest the existence of a bordism version of the Zeeman duality spectral sequence [1].