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Hegenbarth, Friedrich; Repovš, Dušan

Applications of controlled surgery in dimension 4: examples. (English) Zbl 1125.57015

J. Math. Soc. Japan 58, No. 4, 1151-1162 (2006).
Let \(M\) be the complement of a tubular neighborhood of a torus knot in \(S^3\), let \(B\) be the spine of \(M\), which is a compact \(2\)-dimensional polyhedron onto which \(M\) collapses, and consider the closed \(4\)-dimensional manifold \(X=\partial( M\times D^2)\). A main construction of this paper is a direct geometric verification that the natural map \(p: X\to B\) is \(UV^1\).
The authors show how this result, together with E. K. Pedersen, F. Quinn, and A. Ranicki [Controlled surgery with trivial local fundamental groups. High-dimensional manifold topology. Proceedings of the school, ICTP, Trieste, Italy, May 21–June 8, 2001. River Edge, NJ: World Scientific. 421–426 (2003; Zbl 1050.57025)], implies that surgery theory works for \(X\).
They also verify that if \(Y\) is a \(4\)-dimensional Poincaré complex with \(\pi_1Y=\pi_1F\) for some closed oriented aspherical surface \(F\) and the intersection form of \(Y\) is extended from \(\mathbb{Z}\), then \(Y\) is homotopy equivalent to a closed manifold \(M\) for which there is a \(UV^1\) map \(M\to F\). They conclude that surgery theory works for \(Y\).
Reviewer: Bruce Hughes (Nashville)

Cited in 2 Documents

MSC:

57R67 Surgery obstructions, Wall groups
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)

Keywords:

controlled surgery; surgery theory; good fundamental group

Citations:

Zbl 1050.57025
PDFBibTeX XMLCite
\textit{F. Hegenbarth} and \textit{D. Repovš}, J. Math. Soc. Japan 58, No. 4, 1151--1162 (2006; Zbl 1125.57015)
Full Text: DOI arXiv Euclid

References:

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