Geometriae Dedicata

, Volume 151, Issue 1, pp 155–173 | Cite as

Polyhedral surfaces in wedge products

  • Thilo Rörig
  • Günter M. ZieglerEmail author
Original Paper


We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated “subdirect product” as introduced by McMullen (Discrete Math 14:347–358, 1976); it is dual to the “wreath product” construction of Joswig and Lutz (J Combinatorial Theor A 110:193–216, 2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces “of unusually large genus” that first appeared in works by Coxeter (Proc London Math Soc 43:33–62, 1937), Ringel (Abh Math Seminar Univ Hamburg 20:10–19, 1956), and McMullen et al. (Israel J Math 46:127–144, 1983). Via “projections of deformed wedge products” we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in \({{\mathbb R}^3}\) . As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (“moduli”) for the surfaces in \({{\mathbb R}^3}\) . In order to prove that there are many moduli, we introduce the concept of “affine support sets” in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in \({{\mathbb R}^3}\) via dual 4-polytopes.


Convex polytopes Polyhedral surfaces Wreath products of polytopes Combinatorially regular polyhedral surfaces Surfaces of “unusually high genus” Moduli 

Mathematics Subject Classification (2000)

51M20 52B70 


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  1. 1.
    Amenta N., Ziegler G.M.: Deformed products and maximal shadows. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds) Advances in Discrete and Computational Geometry (South Hadley, MA 1996), Contemporary Math., vol. 223., pp. 57–90. American Mathematical Society, Providence, RI (1998)Google Scholar
  2. 2.
    Betke U., Gritzmann P.: A combinatorial condition for the existence of polyhedral 2-manifolds. Israel J. Math. 42, 297–299 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bokowski J.: A geometric realization without self-intersections does exist for Dyck’s regular map. Discrete Comput. Geom. 4, 583–589 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brehm U.: Maximally symmetric polyhedral realizations of Dyck’s regular map. Mathematika 34, 229–236 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Coxeter H.S.M.: Regular skew polyhedra in three and four dimensions, and their topological analogues. Proc. London Math. Soc 43, 33–62 (1937)zbMATHCrossRefGoogle Scholar
  6. 6.
    Coxeter H.S.M.: The abstract groups G m,n,p. Trans. Am. Math. Soc. 45, 73–150 (1939)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Coxeter H.S.M., Moser W.O.J.: Generators and Relations for Discrete Groups, fourth edition, Ergebnisse Math. Grenzgebiete, vol 14. Springer, Berlin (1980)Google Scholar
  8. 8.
    Crapo H.: The combinatorial theory of structures, Matroid theory (Szeged, 1982). Colloq. Math. Soc. János Bolyai 40, 107–213 (1985)MathSciNetGoogle Scholar
  9. 9.
    Fischli S., Yavin D.: Which 4-manifolds are toric varieties?. Math. Z. 215, 179–185 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fritzsche, K., Holt, F.B.: More polytopes meeting the conjectured Hirsch bound. Discrete Math. 205 (1999)Google Scholar
  11. 11.
    Grünbaum, B.: Convex Polytopes, Graduate Texts in Math., vol. 221. Springer, New York. [Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler (original edition: Interscience, London 1967)] (2003)Google Scholar
  12. 12.
    Joswig M., Lutz F.H.: One-point suspensions and wreath products of polytopes and spheres. J. Combinatorial Theor. Ser. A 110, 193–216 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Joswig M., Rörig T.: Neighborly cubical polytopes and spheres. Israel J. Math. 159, 221–242 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Joswig, M., Ziegler, G.M.: Neighborly cubical polytopes. Discrete Comput. Geom. [Grünbaum Festschrift: Kalai, G., Klee, V. (eds.)] 24: 325–344 (2000)Google Scholar
  15. 15.
    MacPherson, R.D.: Equivariant invariants and linear graphs, “Geometric Combinatorics”. In: Miller, E., Reiner, V., Sturmfels, B. (eds.) Procedings of Park City Mathematical Institute (PCMI) 2004 (Providence, RI), American Mathematical Society, pp. 317–388 (2007)Google Scholar
  16. 16.
    McMullen P.: Constructions for projectively unique polytopes. Discrete Math. 14, 347–358 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    McMullen P., Schulte E., Wills J.M.: Infinite series of combinatorially regular polyhedra in three-space. Geom. Dedicata 26, 299–307 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    McMullen P., Schulz C., Wills J.M.: Equivelar polyhedral manifolds in E 3. Israel J. Math. 41, 331–346 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    McMullen P., Schulz C., Wills J.M.: Polyhedral 2-manifolds in E 3 with unusually large genus. Israel J. Math. 46, 127–144 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ringel G.: Über drei kombinatorische Probleme am n-dimensionalen W ürfel und W ürfelgitter. Abh. Math. Seminar Univ. Hamburg 20, 10–19 (1956)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rörig, T.: Polyhedral surfaces, polytopes, and projections, Ph.D. thesis, TU Berlin (2008). Published online
  22. 22.
    Rörig T., Sanyal R.: Non-projectability of polytope skeleta. Jahresbericht der DMV 112(2), 73–98 (2010)Google Scholar
  23. 23.
    Santos, F., Kim, E.D.: An update on the Hirsch conjecture: Fifty-two years later, preprint, arXiv:, 2009
  24. 24.
    Sanyal, R., Ziegler, G.M.: Construction and analysis of projected deformed products. Discrete Comput. Geom. 43, 412–435 (2010).
  25. 25.
    Ziegler G.M.: Projected products of polygons. Electron. Res. Announce. AMS 10, 122–134 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ziegler G.M.: Projected polytopes, Gale diagrams, and polyhedral surfaces (joint work with Raman Sanyal and Thilo Schröder). Oberwolfach Rep. 2, 986–989 (2005)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Inst. Mathematics MA 8-3TU BerlinBerlinGermany
  2. 2.Inst. Mathematics MA 6-2TU BerlinBerlinGermany

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