×

Characterization of pseudo-collarable manifolds with boundary. (English) Zbl 1471.57024

The paper under review represents a continuation of the recent work by C. Guilbault and the author [J. Topol. Anal. 12, No. 4, 1073–1101 (2020; Zbl 1460.57029)] concerning the problem of finding an appropriate boundary for non-compact manifolds, with the aim of extending Siebenmann’s celebrated Ph.D. Thesis results.
Siebenmann’s old theorem gives necessary and sufficient conditions for a smooth, open, high-dimensional manifold to be compactifiable (i.e. the interior of a smooth compact manifold with boundary). Its natural generalization deals with manifolds with non-compact boundaries: in such a case one looks for a completion of the manifold (i.e. a compact manifold \(\hat M\) and a compactum \(C\) inside \(\partial \hat M\) such that \(\hat M - C\) is homeomorphic to the original manifold \(M\)).
On the other hand, the original problem of Siebenmann can be viewed also as the following question: “When does an open \(n\)-manifold contain an ‘open collar’ neighborhood of infinity?” (A manifold \(M\) is an open collar if \(M \simeq \partial M \times [0, 1)\)). Hence, one can generalize the notion of an open collar to that of a “pseudo-collar”, and then seek conditions that imply that a given open \(n\)-manifold contains a pseudo-collar neighborhood of infinity. Here a manifold \(V\) with compact boundary is a homotopy collar if \(\partial V \to V\) is a homotopy equivalence. If, in addition, \(V\) contains arbitrarily small homotopy collar neighborhoods of infinity, \(V\) is called a pseudo-collar.
In [loc. cit.], Guilbault and the author focused just on the first approach, manifolds with non compact boundaries, and characterized these, by proving that a high-dimensional manifold admits a completion if and only if it satisfies four properties concerning the topology of its neighborhoods of infinity.
However, it turns out that one of these four properties (which concerns the algebraic behaviour of the fundamental group at infinity) is far too strong in order to catch some exotic examples of universal covering spaces which arise somehow naturally in the context of the topology at infinity of discrete groups (e.g. M. W. Davis’ famous examples of universal covers that are not simply connected at infinity [Ann. Math. (2) 117, 293–324 (1983; Zbl 0531.57041)]).
Davis’ open manifolds are actually examples of pseudo-collarable manifolds that are not collarable. In the 2000’s Giulbault carried out a project for a generalization of Siebenmann’s result for pseudo-collarable manifolds with compact boundary, which he completed in 2006 (together with F. Tinsley) providing a complete characterization of these manifolds in [C. R. Guilbault and F. C. Tinsley, Geom. Topol. 10, 541-556 (2006; Zbl 1130.57032)].
The present paper presents a generalization of this result in the context of manifolds with noncompact boundaries, and it furnishes a complete characterization of high-dimensional (\(n\geq 6\)) pseudo collarable open manifolds (possibly with non-compact boundaries).

MSC:

57N99 Topological manifolds
57Q12 Wall finiteness obstruction for CW-complexes
57R65 Surgery and handlebodies
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] [CS76] T. A. Chapman and L. C. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976), no. 3-4, 171-208. Zentralblatt MATH: 0361.57008
Digital Object Identifier: doi:10.1007/BF02392417
Project Euclid: euclid.acta/1485889935
· Zbl 0361.57008
[2] [Dav83] M. W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. 117 (1983), 293-325. Zentralblatt MATH: 0531.57041
Digital Object Identifier: doi:10.2307/2007079
· Zbl 0531.57041
[3] [FQ90] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Math. Ser., 39, Princeton University Press, Princeton, NJ, 1990. Zentralblatt MATH: 0705.57001
· Zbl 0705.57001
[4] [Geo08] R. Geoghegan, Topological methods in group theory, Grad. Texts in Math., 243, Springer, New York, 2008. Zentralblatt MATH: 1141.57001
· Zbl 1141.57001
[5] [Gu] S. Gu, Contractible open manifolds which embed in no compact, locally connected and locally \(1\)-connected metric space, preprint, arXiv:1809.02628. arXiv: 1809.02628
[6] [GG] S. Gu and C. R. Guilbault, Compactifications of manifolds with boundary, J. Topol. Anal. (to appear), arXiv:1712.05995. arXiv: 1712.05995
[7] [Gui00] C. R. Guilbault, Manifolds with non-stable fundamental groups at infinity, Geom. Topol. 4 (2000), 537-579. Zentralblatt MATH: 0958.57023
Digital Object Identifier: doi:10.2140/gt.2000.4.537
Project Euclid: euclid.gt/1513883296
· Zbl 0958.57023
[8] [Gui01] C. R. Guilbault, A non-\( \mathcal{Z} \)-compactifiable polyhedron whose product with the Hilbert cube is Z-compactifiable, Fund. Math. 168 (2001), no. 2, 165-197. · Zbl 0988.57012
[9] [Gui16] C. R. Guilbault, Ends, shapes, and boundaries in manifold topology and geometric group theory, Topology and geometric group theory, Springer Proc. Math. Stat., 184, pp. 45-125, Springer, Cham, 2016. Zentralblatt MATH: 1434.57019
· Zbl 1434.57019
[10] [GT03] C. R. Guilbault and F. C. Tinsley, Manifolds with non-stable fundamental groups at infinity. II, Geom. Topol. 7 (2003), 255-286. Zentralblatt MATH: 1032.57020
Digital Object Identifier: doi:10.2140/gt.2003.7.255
Project Euclid: euclid.gt/1513883098
· Zbl 1032.57020
[11] [GT06] C. R. Guilbault and F. C. Tinsley, Manifolds with non-stable fundamental groups at infinity. III, Geom. Topol. 10 (2006), 541-556. Zentralblatt MATH: 1130.57032
Digital Object Identifier: doi:10.2140/gt.2006.10.541
Project Euclid: euclid.gt/1513799714
· Zbl 1130.57032
[12] [Hat02] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. Zentralblatt MATH: 1044.55001
· Zbl 1044.55001
[13] [KM62] J. M. Kister and D. R. Jr. McMillan, Locally Euclidean factors of \(\mathbb{E}^4\) which cannot be embedded in \(\mathbb{E}^3 \), Ann. of Math. 76 (1962), 541-546. · Zbl 0115.40703
[14] [O’B83] G. O’Brien, The missing boundary problem for smooth manifolds of dimension greater than or equal to six, Topology Appl. 16 (1983), no. 3, 303-324. Zentralblatt MATH: 0567.57007
Digital Object Identifier: doi:10.1016/0166-8641(83)90027-5
· Zbl 0567.57007
[15] [Qui71] D. Quillen, Cohomology of groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 47-51, Gauthier-Villars, Paris, 1971. Zentralblatt MATH: 0225.18011
· Zbl 0225.18011
[16] [RS82] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, Berlin, 1982, Springer Study Edition, Reprint. Zentralblatt MATH: 0477.57003
· Zbl 0477.57003
[17] [Sie65] L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, ProQuest LLC, Ann Arbor, MI, 1965, Ph.D. Thesis, Princeton University.
[18] [Ste] R. W. Sternfeld, A contractible open \(n\)-manifold that embeds in no compact \(n\)-manifold, ProQuest LLC, Ann Arbor, MI, 1977, Ph.D. Thesis, The University of Wisconsin-Madison.
[19] [Wal65] C. · Zbl 0152.21902
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.