Controlled simple homotopy theory and applications
The simple homotopy theory, invented by J. H. C. Whitehead, has found many applications. Today's reference on this subject is the excellent textbook by \textit{M. M. Cohen} [A course in simple-homotopy theory (1973; Zbl 0261.57009)]. This booklet brings us under the name of ''controlled'' simple homotopy theory'' a generalization of the ''classical'' simple homotopy theory which may now be attributed as ''uncontrolled''. The controlled setting comes from the notion of ''controlled homotopy'' in a space which means the following: Let p: \(A\to B\) be a map of a space A into a metric space B. Then a \(p^{-1}(\epsilon)\)-homotopy in A is a homotopy for which the p-image of the path of each point has the diameter \(<\epsilon\) in B. Now, let X and Y be polyhedra and p:\(Y\to B\) be a map (controlling map) into a metric space B (the parameter space). A map f: \(X\to Y\) is a \(p^{-1}(\epsilon)\)-equivalence provided there exists a map g: \(Y\to X\) for which there are homotopies \(fg\simeq id_ Y\), \(gf\simeq id_ X\) such that \(fg\simeq id_ Y\) is a \(p^{-1}(\epsilon)\)-homotopy in Y and that the composition of the homotopy \(gf\simeq id_ X\) with f gives a \(p^{-1}(\epsilon)\) homotopy in Y. This is the notion of ''controlled homotopy equivalence''. It turnes out that the following equivalent notion of the definition of simple homotopy equivalence is convenient for this generalization: f: \(X\to Y\) is a simple homotopy equivalence if there exist a compact polyhedron Z and CE-PL surjections r: \(Z\to X\) and s: \(Z\to Y\) such that fr is homotopic to s. Now, its controlled version reads as follows: A map f: \(X\to Y\) is \(p^{-1}(\epsilon)\)-simple provided there exist a compact polyhedron Z and CE-PL maps r:\(Z\to X\), s:\(Z\to Y\) such that fr and s are \(p^{- 1}(\epsilon)\)-homotopic in Y. These are two basic definitions of the controlled simple homotopy theory which is merely the obstruction theory that arises when one attempts to answer the analogue of the basic problem of the simple homotopy theory, i.e. the following question: For given \(\epsilon >0\), which \(p^{-1}(\delta)\)-equivalences f:\(X\to Y\) are \(p^{-1}(\epsilon)\)-simple? Taking \(B=\{point\}\) one sees that the above notions reduce to the classical (uncontrolled) ones. The same happens with the controlled Whitehead group which is introduced next. Having defined controlled Whitehead torsion the author has all the notions needed to develop this more general theory. Almost all results, together with the main applications of the simple homotopy theory, are generalized in the monograph. For the purpose of getting information on results and applications that are generalized we list the contents: O. Introduction. 1. Applications. 2. Definitions and notation. 3. Construction of \(Wh(Y)_{\epsilon}\). 4. Functorial properties. 5. Controlled Whitehead torsion. 6. Construction of \(K_ 0(Y)_{\epsilon}\). 7. Controlled finiteness obstruction. 8. Further properties of the controlled finiteness obstruction. 9. The splitting homomorphism. 10. The splitting sequence. 11. The realization theorem. 12. Calculations. 13. The controlled boundary theorem. 14. The controlled s-cobordism theorem. The whole controlled simple homotopy theory is not just a plain generalization of the classical one. Namely, there have already been constructed controlled obstructions which now fall under this general theory. One may expect even further deep applications of this promising theory.
scientific article; zbMATH DE number 3874224