A boundedly controlled finiteness obstruction

Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)




Douglas R. Anderson


Topology, Boundedly controlled, Finiteness obstruction

Subject Categories

Geometry and Topology | Mathematics


A CW complex X is finitely dominated if there exists a finite CW complex Y together with continuous maps [Special characters omitted.] such that [Special characters omitted.]

C.T.C. Wall asked the following question, "If X is finitely dominated, does X necessarily have the homotopy type of some finite CW complex"? He went on to discover the answer in general is, "No". In 1965 in [Wa], a now classic paper, he proved the following theorem:

Theorem: Suppose X is finitely dominated. Then there exists an invariant w ( X ) ∈ K 0 ( Z π( X )) such that X has the homotopy type of a finite CW complex if and only if w ( X ) = 0.

Here K 0 ( Z π( X )) is the reduced projective class group of the integral group ring Z π( X ) of the fundamental group of X . The invariant w ( X ) is called Wall's finiteness obstruction for X .

In the 1980's, Douglas R. Anderson and Hans J. Munkholm developed a new theory in the general area of 'topology with control' or spaces 'parametrized over a space' called boundedly controlled ( bc ) topology . The geometry and algebraic topology of be spaces was introduced in [AM 1].

In this dissertation, we review the fundamentals of boundedly controlled topology and generalize Wall's finiteness obstruction theorem to the category of boundedly controlled CW complexes.


Surface provides description only. Full text is available to ProQuest subscribers. Ask your Librarian for assistance.