×

On a question of R.H. Bing concerning the fixed point property for two-dimensional polyhedra. (English) Zbl 1354.55002

This article is motivated by a question of R.H. Bing on the existence of compact 2-dimensional polyhedra \(X\) with the fixed point property and even Euler characteristic. The problem is still open if the condition on \(\chi(X)\) is replaced by \(\widetilde{H}_{*}(X;\mathbb{Q}) \neq 0\). Assuming such \(X\) exists, the authors prove certain restrictions on the fundamental group \(\pi_1(X)\).
Recall that \(X\) satisfies the fixed point property if every continuous map \(f: X \to X\) has a fixed point. Compact 2-dimensional polyhedra satisfying the fixed point property and \(\widetilde{H}_{*}(X;\mathbb{Q}) \neq 0\) are called Bing spaces throughout the article. The main theorem shows that there are no Bing spaces with abelian \(\pi_1(X)\). Moreover, the authors prove that the fundamental group of Bing spaces has non-trivial Schur multiplier, i.e., \(H_2(\pi_1(X)) \neq 0\) and also show that \(\pi_1(X)\) is not isomorphic to any dihedral group, the alternating groups \(A_4, A_5\) or the symmetric group \(S_4\).

MSC:

55M20 Fixed points and coincidences in algebraic topology
57M20 Two-dimensional complexes (manifolds) (MSC2010)
55P15 Classification of homotopy type
57M05 Fundamental group, presentations, free differential calculus
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bing, R. H., The elusive fixed point property, Amer. Math. Monthly, 76, 119-132 (1969) · Zbl 0174.25902
[2] Brown, R. F., The Lefschetz Fixed Point Theorem (1971), Scott, Foresman and Co.: Scott, Foresman and Co. Glenview, Ill.-London, vi+186 pp · Zbl 0216.19601
[3] Brown, K. S., Cohomology of Groups, Graduate Texts in Mathematics, vol. 87 (1982), Springer-Verlag: Springer-Verlag New York-Berlin, x+306 pp
[4] Dold, A., Lectures on Algebraic Topology (1980), Springer-Verlag: Springer-Verlag Berlin-New York, xi+377 pp · Zbl 0234.55001
[5] Gorenstein, D.; Lyons, R.; Solomon, R., The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple K-groups, Mathematical Surveys and Monographs, vol. 40.3 (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, xvi+419 pp · Zbl 0890.20012
[6] Gutierrez, M.; Latiolais, M. P., Partial homotopy type of finite two-complexes, Math. Z., 207, 359-378 (1991) · Zbl 0712.55007
[7] Hagopian, C. L., An update on the elusive fixed-point property, (Pearl, E., Open Problems in Topology. II (2007), Elsevier B. V.), 263-277
[8] Hambleton, I.; Kreck, M., Cancellation of lattices and finite two-complexes, J. Reine Angew. Math., 442, 91-109 (1993) · Zbl 0779.57002
[9] (Hog-Angeloni, C.; Metzler, W.; Sieradski, A. J., Two Dimensional Homotopy and Combinatorial Group Theory. Two Dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Series, vol. 197 (1993), Cambridge University Press: Cambridge University Press Cambridge), xii+412 pp · Zbl 0788.00031
[10] Jiang, B., On the least number of fixed points, Amer. J. Math., 102, 749-763 (1980) · Zbl 0455.55001
[11] Jiang, B., Lectures on Nielsen Fixed Point Theory, Contemp. Math., vol. 14 (1983), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, vii+110 pp · Zbl 0512.55003
[12] Lopez, W., An example in the fixed point theory of polyhedra, Bull. Amer. Math. Soc., 73, 922-924 (1967) · Zbl 0158.41902
[13] Spanier, E., Algebraic Topology (1966), McGraw-Hill Book Co.: McGraw-Hill Book Co. New York-Toronto, Ont.-London, xiv+528 pp · Zbl 0145.43303
[14] Waggoner, R., A fixed point theorem for \((n - 2)\)-connected \(n\)-polyhedra, Proc. Amer. Math. Soc., 33, 143-145 (1972) · Zbl 0212.55601
[15] Waggoner, R., A method of combining fixed points, Proc. Amer. Math. Soc., 51, 191-197 (1975) · Zbl 0307.55003
[16] Weibel, C. A., An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (1994), Cambridge University Press: Cambridge University Press Cambridge, xiv+450 pp · Zbl 0797.18001
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.