Journal of Homotopy and Related Structures

, Volume 14, Issue 2, pp 371–391 | Cite as

Topology of scrambled simplices

  • Dmitry N. KozlovEmail author


In this paper we define a family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a \(\Delta \)-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a large family of both \(\Delta \)-complexes, as well as simplicial complexes, which are contractible, but not collapsible.


\(\Delta \)-Complexes Triangulated spaces Trisps Collapsible Dunce hat 


  1. 1.
    Andersen, R.N., Marjanović, M.M., Schori, R.M.: Symmetric products and higher-dimensional dunce hats. Topol. Proc. 18, 7–17 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Borsuk, K., Ulam, S.: On symmetric products of topological spaces. Bull. AMS 37, 235–244 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cohen, M.: A course in simple-homotopy theory, graduate texts in mathematics, vol. 10, p. x+144. Springer, New York, Berlin (1973)CrossRefGoogle Scholar
  4. 4.
    Eilenberg, S., Zilber, J.A.: Semi-simplicial complexes and singular homology. Ann. of Math. (2) 51, 499–513 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Forman, R.: Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gelfand, S.I., Manin, Y.I.: Methods of homological algebra, Springer monographs in mathematics, 2nd edn, p. xx+372. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hatcher, A.: Algebraic topology, p. xii+544. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  8. 8.
    Hu, S.-T.: The homotopy addition theorem. Ann. of Math. (2) 58, 108–122 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kozlov, D.N.: Collapsibility of \(\Delta (\Pi_n)/S_n\) and some related CW complexes. Proc. AMS 128, 2253–2259 (2000)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kozlov, D.N.: Rational homology of spaces of complex monic polynomials with multiple roots. Mathematika 49(1–2), 77–91 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kozlov, D.N.: Discrete Morse theory for free chain complexes. C. R. Math. Acad. Sci. Paris 340(12), 867–872 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kozlov, D.N.: Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics 21. Berlin Heidelberg: Springer. XX, 390 pp. 115 illus (2008)Google Scholar
  13. 13.
    Spanier, E.H.: Algebraic topology, corrected reprint of the 1966 original, p. xvi+528. Springer, New York (1966)Google Scholar
  14. 14.
    Whitehead, G.W.: Elements of homotopy theory, graduate texts in mathematics 61. Springer, New York, Berlin (1978)CrossRefGoogle Scholar
  15. 15.
    Zeeman, E.C.: On the dunce hat. Topology 2, 341–358 (1964)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BremenBremenGermany

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