Skip to main content

Polyhedral surfaces in wedge products

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4

    Brehm U.: Maximally symmetric polyhedral realizations of Dyck’s regular map. Mathematika 34, 229–236 (1987)

    MathSciNet  MATH  Article  Google 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)

    MATH  Article  Google Scholar 

  6. 6

    Coxeter H.S.M.: The abstract groups G m,n,p. Trans. Am. Math. Soc. 45, 73–150 (1939)

    MathSciNet  MATH  Google 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)

    MathSciNet  Google Scholar 

  9. 9

    Fischli S., Yavin D.: Which 4-manifolds are toric varieties?. Math. Z. 215, 179–185 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10

    Fritzsche, K., Holt, F.B.: More polytopes meeting the conjectured Hirsch bound. Discrete Math. 205 (1999)

  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)

  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)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13

    Joswig M., Rörig T.: Neighborly cubical polytopes and spheres. Israel J. Math. 159, 221–242 (2007)

    MathSciNet  MATH  Article  Google 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)

  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)

  16. 16

    McMullen P.: Constructions for projectively unique polytopes. Discrete Math. 14, 347–358 (1976)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18

    McMullen P., Schulz C., Wills J.M.: Equivelar polyhedral manifolds in E 3. Israel J. Math. 41, 331–346 (1982)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  21. 21

    Rörig, T.: Polyhedral surfaces, polytopes, and projections, Ph.D. thesis, TU Berlin (2008). Published online http://opus.kobv.de/tuberlin/volltexte/2009/2145/

  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: http://arxiv.org/abs/0907.1186, 2009

  24. 24

    Sanyal, R., Ziegler, G.M.: Construction and analysis of projected deformed products. Discrete Comput. Geom. 43, 412–435 (2010). http://arxiv.org/abs/0710.2162

  25. 25

    Ziegler G.M.: Projected products of polygons. Electron. Res. Announce. AMS 10, 122–134 (2004)

    MathSciNet  MATH  Article  Google 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 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Günter M. Ziegler.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rörig, T., Ziegler, G.M. Polyhedral surfaces in wedge products. Geom Dedicata 151, 155–173 (2011). https://doi.org/10.1007/s10711-010-9524-5

Download citation

Keywords

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

Mathematics Subject Classification (2000)

  • 51M20
  • 52B70