Boundedly controlled topology. Foundations of algebraic topology and simple homotopy theory
The theory of spaces parametrized over a space with control is the result of a new direction of research in algebraic and geometric topology. One aspect of this theory, the topology of \(\epsilon\)-control, has been initiated by \textit{F. Quinn} [Ann. Math., II. Ser. 110, 275-331 (1979; Zbl 0413.57012); Invent. Math. 68, 353-424 (1982; Zbl 0533.57008)] and \textit{T. A. Chapman} [Controlled simple homotopy theory and applications (Lect. Notes Math. 1009) (1983; Zbl 0548.57001)]. The second aspect, that of boundedly controlled topology, forms the subject of the present work. A boundedness control structure on a space Z is defined as a pair (P,C), where P is a class of subspaces of Z and \(C: P\to P\) is an order preserving function, satisfying certain inclusion, bounding, and covering properties. The pair (P,C) defines a category \({\mathcal P}\), whose abstraction leads to a general concept of ``category with endomorphism''. A boundedly controlled space (bc space) over a space Z with boundedness control structure is a pair (X,p), where \(p: X\to Z\) is a continuous map. This definition generates a category \({\mathcal T}op^ c/Z\) of bc spaces over Z; similarly, for CW complexes and finite CW complexes, the categories \({\mathcal C}{\mathcal W}^ c/Z\) and \({\mathcal C}{\mathcal W}^ c_ f/Z\) are obtained. In Chapter I the authors develop the category theoretic tools needed for the study of the category \({\mathcal T}op^ c/Z\). The focus is on certain categories of fractions \({\mathcal B}(\Sigma^{-1})\), where \({\mathcal B}\) is a category with endomorphism and \(\Sigma\) is a naturally defined family of morphisms in \({\mathcal B}\). An especially important case is that when \({\mathcal B}\) is a category of functors. The motivation for this work lies in the fact that most algebraic topological invariants of the spaces in \(Top^ c/Z\) are constructed in these categories of fractions. Chapter II is devoted to the algebraic topology of bc spaces. First the idea of a fragmented space and of a fragmentation of a bc space or bc CW complex is introduced. This defines a category \({\mathcal F}{\mathcal T}op/{\mathcal P}\) of fragmented spaces and a fragmentation functor Fr: \({\mathcal T}op^ c/Z\to {\mathcal F}{\mathcal T}op/{\mathcal P}\). Then the fundamental groupoid \(G_ 1(X,p)\), the homology invariants, and the main homotopy invariants, including the analogues of the usual homotopy groups, are defined; the analogues of the absolute and relative Hurewicz theorems are proved. By composing these invariants with the fragmentation functor Fr, the homology and homotopy invariants of bc spaces and pairs are defined and the appropriate Hurewicz theorems are derived. For bc maps between bc CW complexes, a cellular approximation theorem and two versions of the Whitehead theorem (one involving homotopy and one involving homology) are proved. Chapter III deals with the foundations of simple homotopy theory in the category \({\mathcal C}{\mathcal W}^ c_ f/Z\). The appropriate notions of elementary expansions and collapses are introduced and used to define the bc geometric Whitehead group \(WH^ c(X,p)\). Then \(Wh^ c\) is shown to be a homotopy functor, and a 2-index theorem is established. Starting with a ring R and a functor \(G: {\mathcal P}\to groupoids\), the authors define in Chapter IV the category of R\({\mathcal P}G\) modules; this allows them to define the groups \(K_ 1(R{\mathcal P}G)\) and Wh(R\({\mathcal P}G)\) which are the bc analogues of the usual \(K_ 1\) and Whitehead groups. As a special case, the algebraic Whitehead group Wh \({\mathbb{Z}}{\mathcal P}G_ 1(X,p)\) of a bc space (X,p) is derived. Then, generalized Stiefel-Whitney classes are defined, which give rise to involutions on \(K_ 1(R{\mathcal P}G)\) and Wh(R\({\mathcal P}G)\). The isomorphism \(\tau\) : Wh\({}^ c(X,p)\to Wh {\mathbb{Z}}{\mathcal P}G_ 1(X,p)\) is established in Chapter V. Following Milnor's approach, the Whitehead torsion for based R\({\mathcal P}G\) modules is defined and its combinatorial invariance is proved. Chapter VI is devoted to the study of the category \({\mathcal P}{\mathcal L}^ c/Z\), which is the bc analogue of the ordinary category of compact PL manifolds. The main results established here are a bc s-cobordism theorem and its companion realization theorem, as well as a duality theorem for the torsion of a bc PL h-cobordism. On the way to these results, various constructions as bc collars, bc handles and handle decompositions, and the geometric connectivity of manifolds, are investigated. The final Chapter VII approaches the description of \(K_ 1R{\mathcal P}G_ 1(X,p)\) and WhR\({\mathcal P}G_ 1(X,p)\) in terms of standard K-theoretic invariants of X. The authors' results are compared to those of \textit{E. K. Pedersen} [J. Algebra 90, 461-475 (1984; Zbl 0545.18003); Lect. Notes Math. 1060, 174-186 (1984; Zbl 0574.55002); Lect. Notes Math. 1217, 306- 320 (1986; Zbl 0613.57014)], \textit{E. K. Pedersen} and \textit{C. Weibel} [Lect. Notes Math. 1126, 166-181 (1984; Zbl 0591.55002)], and \textit{L. Siebenmann} [Indagationes Math. 32, 479-495 (1970; Zbl 0203.560)]. The group Wh \({\mathbb{Z}}{\mathcal P}G_ 1(X,p)\) is calculated explicitly in one special case.
scientific article; zbMATH DE number 193080