Spaces and equations

Walter Taylor

Fundamenta Mathematicae (2000)

  • Volume: 164, Issue: 3, page 193-240
  • ISSN: 0016-2736

Abstract

top
It is proved, for various spaces A, such as a surface of genus 2, a figure-eight, or a sphere of dimension ≠ 1,3,7, and for any set Σ of equations, that Σ cannot be modeled by continuous operations on A unless Σ is undemanding (a form of triviality that is defined in the paper).

How to cite

top

Taylor, Walter. "Spaces and equations." Fundamenta Mathematicae 164.3 (2000): 193-240. <http://eudml.org/doc/212454>.

@article{Taylor2000,
abstract = {It is proved, for various spaces A, such as a surface of genus 2, a figure-eight, or a sphere of dimension ≠ 1,3,7, and for any set Σ of equations, that Σ cannot be modeled by continuous operations on A unless Σ is undemanding (a form of triviality that is defined in the paper).},
author = {Taylor, Walter},
journal = {Fundamenta Mathematicae},
keywords = {topological space; algebraic theory; fundamental group; cohomology ring},
language = {eng},
number = {3},
pages = {193-240},
title = {Spaces and equations},
url = {http://eudml.org/doc/212454},
volume = {164},
year = {2000},
}

TY - JOUR
AU - Taylor, Walter
TI - Spaces and equations
JO - Fundamenta Mathematicae
PY - 2000
VL - 164
IS - 3
SP - 193
EP - 240
AB - It is proved, for various spaces A, such as a surface of genus 2, a figure-eight, or a sphere of dimension ≠ 1,3,7, and for any set Σ of equations, that Σ cannot be modeled by continuous operations on A unless Σ is undemanding (a form of triviality that is defined in the paper).
LA - eng
KW - topological space; algebraic theory; fundamental group; cohomology ring
UR - http://eudml.org/doc/212454
ER -

References

top
  1. [1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20-104. Zbl0096.17404
  2. [2] J. F. Adams and M. F. Atiyah, K-theory and the Hopf invariant, Quart. J. Math. (Oxford) 17 (1966), 31-38. Zbl0136.43903
  3. [3] J. Adem, The relations on Steenrod powers of cohomology classes, in: Algebraic Geometry and Topology, Symposium in honor of Solomon Lefschetz, Princeton Univ. Press, 1957, 191-238. Zbl0199.26104
  4. [4] A. Bateson, Fundamental groups of topological R-modules, Trans. Amer. Math. Soc. 270 (1982), 525-536. Zbl0485.22004
  5. [5] R. Brown, From groups to groupoids: a brief survey, Bull. London Math. Soc. 19 (1987), 113-134. Zbl0612.20032
  6. [6] R. Brown, Topology: a Geometric Account of General Topology, Homotopy Types, and the Fundamental Groupoids, Halsted Press, New York, 1988. Zbl0655.55001
  7. [7] J. P. Coleman, Topologies on free algebras, Ph.D. Thesis, Univ. of Colorado, Boulder, 1992. 
  8. [8] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241-249. Zbl0158.41503
  9. [9] J. Dieudonné, A History of Algebraic and Differential Topology 1900-1960, Birkhäuser, Boston, 1989. Zbl0673.55002
  10. [10] B. Eckmann, T. Ganea and P. Hilton, Generalized means, in: D. Gilbarg et al. (eds.), Studies in Mathematical Analysis, Stanford Univ. Press, 1963, 82-92. Zbl0114.39501
  11. [11] T. Evans, Products of points - some simple algebras and their identities, Amer. Math. Monthly 74 (1976), 363-372. Zbl0207.32901
  12. [12] O. C. García and W. Taylor, The lattice of interpretability types of varieties, Mem. Amer. Math. Soc. 305 (1984). Zbl0559.08003
  13. [13] M. Hall, Jr., The Theory of Groups, Macmillan, New York, 1959. 
  14. [14] P. J. Higgins, Categories and Groupoids, van Nostrand Reinhold, London, 1971. Zbl0226.20054
  15. [15] P. J. Hilton and S. Wylie, Homology Theory, Cambridge Univ. Press, 1960. 
  16. [16] H. Hopf, Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension, Fund. Math. 25 (1935), 427-440. Zbl0012.31902
  17. [17] S.-T. Hu, Homotopy Theory, Academic Press, New York, 1959. 
  18. [18] I. M. James, Multiplications on spheres, I, II, Proc. Amer. Math. Soc. 8 (1957), 192-196, and Trans. Amer. Math. Soc. 84 (1957), 545-558. 
  19. [19] F. W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869-872. Zbl0119.25901
  20. [20] F. W. Lawvere, Some algebraic problems in the context of functorial semantics of algebraic theories, in: Lecture Notes in Math. 61, Springer, 1968, 41-46. 
  21. [21] J.-L. Loday, J. D. Stasheff and A. A. Voronov (eds.), Operads: Proceedings of Renaissance Conferences, Contemp. Math. 202, Amer. Math. Soc., 1997. 
  22. [22] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Interscience, New York, 1966. 
  23. [23] R. N. McKenzie, G. F. McNulty and W. F. Taylor, Algebras, Lattices, Varieties, Volume 1, Wadsworth and Brooks-Cole, Monterey, CA, 1987. 
  24. [24] R. McKenzie and S. Świerczkowski, Non-covering in the interpretability lattice of equational theories, Algebra Universalis 30 (1993), 157-170. Zbl0802.08004
  25. [25] R. McKenzie and W. Taylor, Interpretation of module varieties, J. Algebra 135 (1990), 456-493. Zbl0729.08005
  26. [26] J. R. Munkres, Elements of Algebraic Topology, Benjamin-Cummings, Menlo Park, CA, 1984. Zbl0673.55001
  27. [27] J. Mycielski and W. Taylor, Remarks and problems on a lattice of equational chapters, Algebra Universalis 23 (1986), 24-31. Zbl0616.08014
  28. [28] W. D. Neumann, Representing varieties of algebras by algebras, J. Austral. Math. Soc. 11 (1970), 1-8. Zbl0199.32702
  29. [29] W. D. Neumann, On Malcev conditions, ibid. 17 (1974), 376-384. Zbl0294.08004
  30. [30] S. P. Novikov, Topology, translated from the Russian by B. Botvinnik and R. G. Burns, Encyclopedia Math. Sci. 12, Springer, Berlin, 1996. 
  31. [31] A. Pultr and V. Trnková, Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, Academia, Prague, 1980. 
  32. [32] J. Sichler and V. Trnková, Isomorphism and elementary equivalence of clone segments, Period. Math. Hungar. 32 (1996), 113-128. Zbl0859.54006
  33. [33] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. 
  34. [34] N. E. Steenrod, Cohomology Operations, Ann. of Math. Stud. 50, Princeton Univ. Press, 1962. 
  35. [35] S. /Swierczkowski, Topologies in free algebras, Proc. London Math. Soc. (3) 14 (1964), 566-576. Zbl0123.09902
  36. [36] W. Taylor, Characterizing Mal'cev conditions, Algebra Universalis 3 (1973), 351-397. Zbl0304.08003
  37. [37] W. Taylor, Varieties obeying homotopy laws, Canad. J. Math. 29 (1977), 498-527. Zbl0357.08004
  38. [38] W. Taylor, Laws obeyed by topological algebras-extending results of Hopf and Adams, J. Pure Appl. Algebra 21 (1981), 75-98. Zbl0485.57019
  39. [39] W. Taylor, The Clone of a Topological Space, Res. Exp. Math. 13, Heldermann, 1986. Zbl0615.54013
  40. [40] W. Taylor, Abstract clone theory, in: Algebras and Orders (Montreal, 1991), I. G. Rosenberg and G. Sabidussi (eds.), Kluwer, 1993, 507-530. Zbl0792.08005
  41. [41] V. Trnková, Semirigid spaces, Trans. Amer. Math. Soc. 343 (1994), 305-325. Zbl0803.54015
  42. [42] V. Trnková, Continuous and uniformly continuous maps of powers of metric spaces, Topology Appl. 63 (1995), 189-200. Zbl0827.54007
  43. [43] V. Trnková, Algebraic theories, clones and their segments, Appl. Categ. Structures 4 (1996) 241-249. Zbl0935.18004
  44. [44] V. Trnková, Representation of algebraic theories and nonexpanding maps, J. Pure Appl. Algebra, to appear. Zbl0946.08001
  45. [45] G. W. Whitehead, Elements of Homotopy Theory, Springer, New York, 1978. Zbl0406.55001
  46. [46] G. Wraith, Algebraic Theories, Lecture Notes Series 22, Matematisk Institut, Universitet Aarhus, 1970. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.