# zbMATH — the first resource for mathematics

$$K_{-i}$$ obstructions to factoring an open manifold. (English) Zbl 0654.57009
The authors give the following abstract which adequately describes the paper: “Let $$M^{n+k}$$ be an open PL manifold of dimension $$n+k>5$$, let X be a finite polyhedron, and suppose $$f: M^{n+k}\to X\times {\mathbb{R}}\quad k$$ is a bounded homotopy equivalence. If $$k\geq 1$$, we use radial engulfing and Siebenmann’s twist-gluing (twist $$=$$ id.) to construct a manifold $$M_ 1$$ with infinite cyclic cover M and a bounded homotopy equivalence $$f_ 1: M_ 1\to X\times S\quad 1\times {\mathbb{R}}^{k-1}.$$ By iterating this construction we obtain a manifold $$M_{k-1}$$ and a bounded homotopy equivalence $$f_{k-1}: M_{k-1}\to X\times S\quad 1\times \cdot \cdot \cdot \times S\quad 1\times {\mathbb{R}}.$$ We show that the Siebenmann obstruction $$\sigma_ 1$$ in $$\tilde K_ 0({\mathbb{Z}}\pi_ 1M_{k-1})$$ to factoring $$M_{k-1}=N_ 1\times {\mathbb{R}}$$ is an element of $$\tilde K_{-k+1}({\mathbb{Z}}\pi_ 1X)$$. If $$\sigma_ 1=0$$, let $$\beta$$ be the automorphism of $$\tilde K_ 0$$ induced by mapping the class of a projective P to the class of the projective $$\bar P$$ consisting of antihomomorphisms from P into the coefficient ring $${\mathbb{Z}}\pi_ 1(X\times S$$ $$1\times \cdot \cdot \cdot \times S$$ 1) (k-2 S 1 factors). Then subsequent obstructions lie in the quotients $$\tilde K_{-1}/image (1+\beta).$$ That is, if $$\sigma_ 1$$ vanishes, there exists a sequence of obstructions $$\sigma_ i$$ in $$\tilde K_{-k+i}({\mathbb{Z}}\pi_ 1X)/image (1+\beta),$$ $$i=2,...,s$$, such that $$M=N_ s\times {\mathbb{R}}\quad s,$$ for some PL manifold $$N_ s$$ if and only if each $$\sigma_ i$$ vanishes.”
Reviewer: H.-J.Munkholm

##### MSC:
 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. 57Q30 Engulfing
Full Text:
##### References:
 [1] Anderson, D.; Hsiang, W.C., The functors K-i and pseudo-isotopies of polyhedra, Ann. math., 105, 2, 201-223, (1977) · Zbl 0406.57030 [2] Bass, H., Algebraic K-theory, (1968), Benjamin New York · Zbl 0174.30302 [3] Bass, H.; Heller, A.; Swan, R.G., The Whitehead group of a polynomial extension, Inst. hautes etudes sci. publ. math., 22, 61-80, (1964) · Zbl 0248.18026 [4] Bing, R.H., Radial engulfing, (), 1-18 · Zbl 0186.57505 [5] Carter, D.W., Localization in lower algebraic K-theory, Comm. algebra, 8, 603-622, (1980) · Zbl 0429.16019 [6] Chapman, T.A., Approximation results in Hilbert cube manifolds, Trans. amer. math. soc, 262, 303-334, (1980) · Zbl 0464.57009 [7] Chapman, T.A., Approximation results in topological manifolds, Mem. amer. math. soc., 34, (1981) · Zbl 0478.57009 [8] Cohen, M., A course in simple homotopy theory, (1970), Springer New York [9] Edwards, R.D.; Kirby, R.C., Deformations of spaces of imbeddings, Ann. math., 93, 63-88, (1971) · Zbl 0214.50303 [10] Farrell, F.T.; Hsiang, W.C., A formula for K1(Rα[T]), () · Zbl 0217.04505 [11] Hudson, J.F.P., Piecewise linear topology, (1969), Benjamin New York · Zbl 0189.54507 [12] C.B. Hughes, Bounded homotopy equivalences of Hilbert cube manifolds, preprint. · Zbl 0527.57008 [13] Kelley, J.L., General topology, (1955), Van Nostrand New York · Zbl 0066.16604 [14] Maclane, S., Homology, (1963), Springer Berlin · Zbl 0133.26502 [15] Milnor, J., Whitehead torsion, Bull. amer. math. soc., 72, 358-426, (1966) · Zbl 0147.23104 [16] E.K. Pedersen, On the functors K-i, preprint. [17] Quinn, F., Ends of maps, II, Invent. math., 68, 353-424, (1982) · Zbl 0533.57008 [18] Rourke, C.P.; Sanderson, B.J., Introduction to piecewise-linear topology, (1972), Springer Berlin · Zbl 0254.57010 [19] Siebenmann, L.C., The obstruction to finding a boundary for an open manifold of dimension greater than five, () · Zbl 0215.24603 [20] Siebenmann, L.C., A total Whitehead torsion obstruction to fibering over the circle, Comment. math. helv., 45, 1-48, (1970) · Zbl 0215.24603 [21] Silvester, J.R., Introduction to algebraic K-theory, (1981), Chapman and Hall London · Zbl 0468.18006 [22] Stallings, J., The piecewise linear structure of Euclidean space, Proc. Cambridge philos. soc., 58, 481-488, (1962) · Zbl 0107.40203 [23] Wall, C.T.C., Finiteness conditions for C.W. complexes II, Proc. royal soc., 295, 129-139, (1966), (Great Britain) Ser A [24] Wright, P., Radial engulfing in codimension three, Duke math. J., 38, 295-298, (1971) · Zbl 0217.20102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.