Generalized manifolds in products of curves
HTML articles powered by AMS MathViewer
- by Akira Koyama, Józef Krasinkiewicz and Stanisław Spież PDF
- Trans. Amer. Math. Soc. 363 (2011), 1509-1532 Request permission
Abstract:
The intent of this article is to distinguish and study some $n$-dimensional compacta (such as weak $n$-manifolds) with respect to embeddability into products of $n$ curves. We show that if $X$ is a locally connected weak $n$-manifold lying in a product of $n$ curves, then $\operatorname {rank} H^{1}(X)\ge n$. If $\operatorname {rank} H^{1}(X)=n$, then $X$ is an $n$-torus. Moreover, if $\operatorname {rank} H^{1}(X)<2n$, then $X$ can be presented as a product of an $m$-torus and a weak $(n-m)$-manifold, where $m\ge 2n-\operatorname {rank} H^{1}(X)$. If $\operatorname {rank} H^{1}(X)<\infty$, then $X$ is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak $2$-manifold $X$ lying in a product of two graphs such that $H^{2}(X)=0$.References
- Karol Borsuk, Über das Phänomen der Unzerlegbarkeit in der Polyedertopologie, Comment. Math. Helv. 8 (1935), no. 1, 142–148 (German). MR 1509522, DOI 10.1007/BF01199551
- —, Theory of Retracts, PWN-Polish Scientific Publishers, Warszawa, 1967.
- K. Borsuk, Remark on the Cartesian product of two $l$-dimensional spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 9, 971–973 (English, with Russian summary). MR 394636
- H. G. Bothe, Eine Einbettung $m$-dimensionaler Mengen in einen $(m+1)$-dimensionalen absoluten Retrakt, Fund. Math. 52 (1963), 209–224 (German). MR 151972, DOI 10.4064/fm-52-2-209-224
- Philip L. Bowers, General position properties satisfied by finite products of dendrites, Trans. Amer. Math. Soc. 288 (1985), no. 2, 739–753. MR 776401, DOI 10.1090/S0002-9947-1985-0776401-5
- Robert Cauty, Sur le plongement des surfaces non orientables dans un produit de deux graphes, Bull. Polish Acad. Sci. Math. 32 (1984), no. 1-2, 121–128 (French, with English and Russian summaries). MR 766989
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886, DOI 10.1515/9781400877492
- Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
- David Gillman, Sergei Matveev, and Dale Rolfsen, Collapsing and reconstruction of manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 35–39. MR 1282753, DOI 10.1090/conm/164/01583
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- W. Hurewicz and H. Wallman, Dimension Theory, Princeton, 1948.
- Ivan Ivanšić and Uroš Milutinović, A universal separable metric space based on the triangular Sierpiński curve, Topology Appl. 120 (2002), no. 1-2, 237–271. In memory of T. Benny Rushing. MR 1895494, DOI 10.1016/S0166-8641(01)00017-7
- Yukihiro Kodama, On embeddings of spaces into ANR and shapes, J. Math. Soc. Japan 27 (1975), no. 4, 533–544. MR 400158, DOI 10.2969/jmsj/02740533
- Yukihiro Kodama and Jin Ono, On two notions of shape for pairs of spaces, General Topology and Appl. 6 (1976), no. 2, 207–225. MR 394554, DOI 10.1016/0016-660X(76)90034-9
- A. Koyama, J. Krasinkiewicz, S. Spiez, Embedding compacta into products of curves, arXiv:0712.3470v1 [math.GT] 20 Dec 2007, 1–71.
- —, On embeddings into products of curves - An algebraic approach, preprint.
- Józef Krasinkiewicz, On approximation of mappings into $1$-manifolds, Bull. Polish Acad. Sci. Math. 44 (1996), no. 4, 431–440. MR 1420956
- J. Krasinkiewicz, On a method of constructing ANR-sets. An application of inverse limits, Fund. Math. 92 (1976), no. 2, 95–112. MR 420546, DOI 10.4064/fm-92-2-95-112
- Włodzimierz Kuperberg, On embeddings of manifolds into Cartesian products of compacta, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 9-10, 845–848 (English, with Russian summary). MR 518991
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
- Stephen Leon Lipscomb, On imbedding finite-dimensional metric spaces, Trans. Amer. Math. Soc. 211 (1975), 143–160. MR 380751, DOI 10.1090/S0002-9947-1975-0380751-8
- William S. Massey, Homology and cohomology theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., New York-Basel, 1978. An approach based on Alexander-Spanier cochains. MR 0488016
- J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999. MR 1702278
- Daria Michalik, Embeddings of $n$-dimensional separable metric spaces into the product of Sierpiński curves, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2661–2664. MR 2302589, DOI 10.1090/S0002-9939-07-08782-5
- John Milnor, On the Steenrod homology theory, Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 79–96. MR 1388297, DOI 10.1017/CBO9780511662676.005
- J. Nagata, Note on dimension theory for metric spaces, Fund. Math. 45 (1958), 143–181. MR 105081, DOI 10.4064/fm-45-1-143-181
- —, Modern Dimension Theory, North-Holland, Amsterdam, 1965.
- Wojciech Olszewski, Embeddings of finite-dimensional spaces into finite products of $1$-dimensional spaces, Topology Appl. 40 (1991), no. 1, 93–99. MR 1114094, DOI 10.1016/0166-8641(91)90061-P
- Roman Pol, A $2$-dimensional compactum in the product of two $1$-dimensional compacta which does not contain any rectangle, Topology Proc. 16 (1991), 133–135. MR 1206460
- Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
- E. G. Sklyarenko, Homology and cohomology theories of general spaces [ MR1004026 (90h:55005)], General topology, II, Encyclopaedia Math. Sci., vol. 50, Springer, Berlin, 1996, pp. 119–256. MR 1392482
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- N. E. Steenrod, Regular cycles of compact metric spaces, Ann. of Math. (2) 41 (1940), 833–851. MR 2544, DOI 10.2307/1968863
- Yaki Sternfeld, Mappings in dendrites and dimension, Houston J. Math. 19 (1993), no. 3, 483–497. MR 1242434
- K\B{o}ichi Tsuda, A note on closed embeddings of finite-dimensional metric spaces, Bull. London Math. Soc. 17 (1985), no. 3, 275–278. MR 806432, DOI 10.1112/blms/17.3.275
- Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095, DOI 10.1090/coll/028
- E. C. Zeeman, On the dunce hat, Topology 2 (1964), 341–358. MR 156351, DOI 10.1016/0040-9383(63)90014-4
Additional Information
- Akira Koyama
- Affiliation: Department of Mathematics, Faculty of Science, Shizuoka University, Suruga, Shizuoka, 422-8529, Japan
- Email: sakoyam@ipc.shizuoka.ac.jp
- Józef Krasinkiewicz
- Affiliation: The Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland
- Email: jokra@impan.pl
- Stanisław Spież
- Affiliation: The Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland
- Email: spiez@impan.pl
- Received by editor(s): April 3, 2008
- Received by editor(s) in revised form: June 4, 2009, June 18, 2009, and July 2, 2009
- Published electronically: October 8, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 1509-1532
- MSC (2010): Primary 54E45, 55N05, 57N35; Secondary 55M10, 57Q05
- DOI: https://doi.org/10.1090/S0002-9947-2010-05157-8
- MathSciNet review: 2737275