Weak variants of Martin's Axiom

J. Barnett

Fundamenta Mathematicae (1992)

  • Volume: 141, Issue: 1, page 61-73
  • ISSN: 0016-2736

Abstract

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Examples exist of smooth maps on the boundary of a smooth manifold M which allow continuous extensions over M without fixed points but no such smooth extensions. Such maps are studied here in more detail. They have a minimal fixed point set when all transversally fixed maps in their homotopy class are considered. Therefore we introduce a Nielsen fixed point theory for transversally fixed maps on smooth manifolds without or with boundary, and use it to calculate the minimum number of fixed points in cases where continuous map extensions behave differently from smooth ones. In the appendix it is shown that a subset of a smooth manifold can be realized as the fixed point set of a smooth map in a given homotopy class if and only if it can be realized as the fixed point set of a continuous one. A special case of this result is used in a proof of the paper.

How to cite

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Barnett, J.. "Weak variants of Martin's Axiom." Fundamenta Mathematicae 141.1 (1992): 61-73. <http://eudml.org/doc/211951>.

@article{Barnett1992,
abstract = {Examples exist of smooth maps on the boundary of a smooth manifold M which allow continuous extensions over M without fixed points but no such smooth extensions. Such maps are studied here in more detail. They have a minimal fixed point set when all transversally fixed maps in their homotopy class are considered. Therefore we introduce a Nielsen fixed point theory for transversally fixed maps on smooth manifolds without or with boundary, and use it to calculate the minimum number of fixed points in cases where continuous map extensions behave differently from smooth ones. In the appendix it is shown that a subset of a smooth manifold can be realized as the fixed point set of a smooth map in a given homotopy class if and only if it can be realized as the fixed point set of a continuous one. A special case of this result is used in a proof of the paper.},
author = {Barnett, J.},
journal = {Fundamenta Mathematicae},
keywords = {consistency; weak variants of Martin's Axiom},
language = {eng},
number = {1},
pages = {61-73},
title = {Weak variants of Martin's Axiom},
url = {http://eudml.org/doc/211951},
volume = {141},
year = {1992},
}

TY - JOUR
AU - Barnett, J.
TI - Weak variants of Martin's Axiom
JO - Fundamenta Mathematicae
PY - 1992
VL - 141
IS - 1
SP - 61
EP - 73
AB - Examples exist of smooth maps on the boundary of a smooth manifold M which allow continuous extensions over M without fixed points but no such smooth extensions. Such maps are studied here in more detail. They have a minimal fixed point set when all transversally fixed maps in their homotopy class are considered. Therefore we introduce a Nielsen fixed point theory for transversally fixed maps on smooth manifolds without or with boundary, and use it to calculate the minimum number of fixed points in cases where continuous map extensions behave differently from smooth ones. In the appendix it is shown that a subset of a smooth manifold can be realized as the fixed point set of a smooth map in a given homotopy class if and only if it can be realized as the fixed point set of a continuous one. A special case of this result is used in a proof of the paper.
LA - eng
KW - consistency; weak variants of Martin's Axiom
UR - http://eudml.org/doc/211951
ER -

References

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  1. [Ba] J. Barnett, Ph.D. thesis, University of Colorado, Boulder 1990. 
  2. [CP] J. Cichoń and J. Pawlikowski, On ideals of subsets of the plane and on Cohen reals, J. Symbolic Logic 51 (1986), 560-569. Zbl0622.03035
  3. [DS] K. Devlin and S. Shelah, A weak version of ⋄ which follows from a weak version of 2 0 ℵ < 2 1 ℵ , Israel J. Math. 29 (1978), 239-247. Zbl0403.03040
  4. [Fr] D. Fremlin, Consequences of Martin's Axiom, Cambridge Tracts in Math. 84, Cambridge University Press, Cambridge 1984. 
  5. [He] C. Herink, Ph.D. thesis, University of Wisconsin, 1977. 
  6. [IS] J. Ihoda and S. Shelah, MA(σ-centered): Cohen reals, strong measure zero sets and strongly meager sets, preprint. 
  7. [KT] K. Kunen and F. Tall, Between Martin's Axiom and Souslin's Hypothesis, Fund. Math. 102 (1979), 173-181. Zbl0415.03040
  8. [Pa] J. Pawlikowski, Finite support iteration and strong measure zero sets, J. Symbolic Logic 55 (1990), 674-677. Zbl0703.03032
  9. [Ro1,2] J. Roitman, Adding a random or a Cohen real: topological consequences and the effect on Martin's axiom, Fund. Math. 103 (1979), 47-60; Correction, ibid. 129 (1988), 141. Zbl0442.03034
  10. [To1] S. Todorčević, Partition Problems in Topology, Contemp. Math. 84, Amer. Math. Soc., Providence 1989. 
  11. [To2] S. Todorčević, Remarks on cellularity in products, Compositio Math. 57 (1986), 357-372. Zbl0616.54002

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