The Michigan Mathematical Journal

A Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifolds

Jeffrey J. Rolland

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Abstract

In this paper, we develop a geometric procedure for producing a “reverse” to Quillen’s plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudocollars, which are stackings of 1-sided h-cobordisms. Each of our pseudocollars has the same boundary and prohomology systems at infinity and similar group-theoretic properties for their profundamental group systems at infinity. In particular, the kernel group of each group extension for each 1-sided h-cobordism in the pseudocollars is the same group. Nevertheless, the profundamental group systems at infinity are all distinct. A good deal of combinatorial group theory is needed to verify this fact, including an application of Thompson’s group V.

The notion of pseudocollars originated in Hilbert cube manifold theory, where it was part of a necessary and sufficient condition for placing a Z-set as the boundary of an open Hilbert cube manifold.

Article information

Source
Michigan Math. J., Volume 67, Issue 3 (2018), 485-509.

Dates
Received: 14 October 2016
Revised: 4 October 2017
First available in Project Euclid: 6 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1522980163

Digital Object Identifier
doi:10.1307/mmj/1522980163

Mathematical Reviews number (MathSciNet)
MR3835562

Zentralblatt MATH identifier
06969982

Subjects
Primary: 57R65: Surgery and handlebodies 57R19: Algebraic topology on manifolds
Secondary: 57S30: Discontinuous groups of transformations 57M07: Topological methods in group theory

Citation

Rolland, Jeffrey J. A Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifolds. Michigan Math. J. 67 (2018), no. 3, 485--509. doi:10.1307/mmj/1522980163. https://projecteuclid.org/euclid.mmj/1522980163


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References

  • [1] F. Ancel and C. Guilbault, $\mathcal{Z}$-Compactifications of open manifolds, Topology 38 (1999), 1265–1280.
  • [2] R. Baer and F. Levi, Freie Produkte und ihre Untergruppen, Compos. Math. 3 (1936), 391–398.
  • [3] K. Brown, The geometry of finitely presented simple groups, Algorithms and classification in combinatorial group theory, Math. Sci. Res. Inst. Publ., pp. 121–136, 1992.
  • [4] M. Brown, Locally flat embeddings of topological manifolds, Ann. of Math. 75 (1962), 331–341.
  • [5] T. Chapman and L. Siebenmann, Finding a boundary for a Hilbert cube manifold, Topology 3 (1965), 171–208.
  • [6] M. Cohen, A course in simple-homotopy theory, first edition, Springer, New York, 1973.
  • [7] M. Curtis and K. Kwun, Infinite sums of manifolds, Acta Math. 137 (1976), 31–42.
  • [8] M. Freedman and F. Quinn, Topology of $4$-manifolds, Princeton University Press, Princeton, 1990.
  • [9] R. Geoghegan, Topological methods in group theory, Springer, New York, 2007.
  • [10] C. Guilbault, Manifolds with non-stable fundamental groups at infinity, Geom. Topol. 4 (2000), 537–579.
  • [11] C. Guilbault, Ends, shapes, and boundaries in manifold topology and geometric group theory, Topology and geometric group theory, Springer Proc. Math. Stat., pp. 45–125, Springer, 2016.
  • [12] C. Guilbault and F. Tinsley, Noncompact manifolds that are inward tame, Pacific J. Math. 288 (2017), no. 1, 87–128.
  • [13] C. Guilbault and F. Tinsley, Manifolds with non-stable fundamental groups at infinity, II, Geom. Topol. 7 (2003), 255–286.
  • [14] C. Guilbault and F. Tinsley, Manifolds with non-stable fundamental groups at infinity, III, Geom. Topol. 10 (2006), 541–556.
  • [15] C. Guilbault and F. Tinsley, Spherical alterations of handles: embedding the manifold plus construction, Algebr. Geom. Topol. 13 (2013), 35–60.
  • [16] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
  • [17] J.-C. Hausmann, Homological surgery, Ann. of Math. 104 (1976), 573–584.
  • [18] M. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67–72.
  • [19] R. Lyndon and P. Schupp, Combinatorial group theory, first edition, Springer, Berlin, 2013.
  • [20] H. Neumann and I. Dey, The Hopf property of free groups, Math. Z. 117 (1970), 325–339.
  • [21] W. Parry, J. Cannon, and W. Floyd, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), 215–256.
  • [22] J. Robbin, Matrix algebra: using MINImal MATlab, first edition, A K Peters, Wellesley, 1994.
  • [23] D. J. S. Robinson, A course in the theory of groups, second edition, Springer, New York, 1995.
  • [24] C. Rourke and B. Sanderson, Introduction to piecewise-linear topology, Springer, Berlin, 1972.
  • [25] L. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. thesis, Princeton, 1965.
  • [26] P. Sparks, Contractible $n$-manifolds and the double $n$-space property, Ph.D. thesis, University of Wisconsin-Milwaukee, 2014.