Buoncristiano, Sandro (1973) Coefficients in bordism. PhD thesis, University of Warwick.

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Abstract

In this paper is given a treatment of coefficients for a vast class of (co)-bordism theories of manifolds with singularities. If h(-) is one of these theories and G is an abelian group we introduce coefficients G into h in such a way that

(a) h(-; G) is a functor on the category of all abelian groups;
(b) the universal-coefficient sequence is natural on the category of all abelian groups. Consequently, by [5], it is pure
and hence, by algebra, it splits for a vast class of abelian groups, including all groups of finite type.
(c) If G is an R-module, R any commutative ring with unit, h(-; G) inherits an R-module and h(point; R)-module structure in a natural way. It follows from Dold [3] that there is a generalized universal-coefficient sequence consisting of a spectral sequence running Torp(hq(-;R),G)=>h(-; G).
(d) When h(-) = H(-;Z) singular (co)homology with integer coefficients, the definition coincides with the usual one given by means of chain complexes.
(e) The method works to introduce local coefficients in the cohomology theory h*(-) ; i.e. if F is any sheaf of modules over a compact polyhedron X, then h*(X; F) is defined and it is a functor on sheaves.
(f) If F/x is a 'nice' sheaf (in the sense of III.3), there is a spectral sequence running

Hp(X; hqF) => h*(X; F) (*)

where hqF is the graded sheaf obtained from F by applying the functor hq(point; -) and H is singular cohomology. When F is constant, (*) reduces to the usual type. A comparison theorem is deduced from (*) by means of the Mapping Theorem between spectral sequences.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Manifolds (Mathematics), Singularities (Mathematics)
Official Date:	1973
Dates:	Date Event 1973 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Rourke, C. P. (Colin Patrick), 1943-
Sponsors:	Consiglio nazionale delle ricerche (Italy)
Extent:	iv, 76 leaves
Language:	eng

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