Simplicial complexes with lattice structures

Bergman, George¹

¹Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-3840, United States

Algebraic and Geometric Topology, Tome 17 (2017) no. 1, pp. 439-486

Résumé

If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex Δ(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L. We note properties of this construction and of some variants, and pose several questions. For M3 the 5–element nondistributive modular lattice, Δ(M3) is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor.

We also describe a construction of “stitching together” a family of lattices along a common chain, and note how Δ(M3) can be regarded as an example of this construction.

Détail

DOI : 10.2140/agt.2017.17.439

Classification : 06B30, 05E45, 06A07, 57Q99
Keywords: order complex of a poset or lattice, topological lattice, distributive lattice, modular lattice, breadth of a lattice

Bibliographie

[1] P Alexandroff, Diskrete Räume, Mat. Sbornik N.S. 2 (1937) 501

[2] L W Anderson, On the distributivity and simple connectivity of plane topological lattices, Trans. Amer. Math. Soc. 91 (1959) 102 | DOI

[3] K A Baker, A R Stralka, Compact, distributive lattices of finite breadth, Pacific J. Math. 34 (1970) 311 | DOI

[4] G M Bergman, Direct limits and fixed point sets, J. Algebra 292 (2005) 592 | DOI

[5] G M Bergman, An invitation to general algebra and universal constructions, Springer (2015) | DOI

[6] T H Choe, The breadth and dimension of a topological lattice, Proc. Amer. Math. Soc. 23 (1969) 82 | DOI

[7] E Dyer, A Shields, Connectivity of topological lattices, Pacific J. Math. 9 (1959) 443 | DOI

[8] D E Edmondson, A nonmodular compact connected topological lattice, Proc. Amer. Math. Soc. 7 (1956) 1157 | DOI

[9] S Eilenberg, N Steenrod, Foundations of algebraic topology, Princeton University Press (1952)

[10] J Folkman, The homology groups of a lattice, J. Math. Mech. 15 (1966) 631

[11] E Fried, G Grätzer, Pasting infinite lattices, J. Austral. Math. Soc. Ser. A 47 (1989) 1 | DOI

[12] E Fried, G Grätzer, E T Schimidt, Multipasting of lattices, Algebra Universalis 30 (1993) 241 | DOI

[13] G Gierz, A R Stralka, Modular lattices on the 3–cell are distributive, Algebra Universalis 26 (1989) 1 | DOI

[14] G Grätzer, Lattice theory: foundation, Springer (2011) | DOI

[15] G Grätzer, F Wehrung, A survey of tensor products and related constructions in two lectures, Algebra Universalis 45 (2001) 117 | DOI

[16] F Harary, Graph theory, Addison-Wesley (1969)

[17] S Mac Lane, Categories for the working mathematician, 5, Springer (1998) | DOI

[18] J J Rotman, An introduction to algebraic topology, 119, Springer (1988) | DOI

[19] W Taylor, Varieties obeying homotopy laws, Canad. J. Math. 29 (1977) 498 | DOI

[20] W Taylor, Possible classification of finite-dimensional compact Hausdorff topological algebras, preprint (2015)

[21] M L Wachs, Poset topology: tools and applications, from: "Geometric combinatorics" (editors E Miller, V Reiner, B Sturmfels), IAS/Park City Math. Ser. 13, Amer. Math. Soc. (2007) 497

[22] J W Walker, Canonical homeomorphisms of posets, European J. Combin. 9 (1988) 97 | DOI

[23] A D Wallace, The structure of topological semigroups, Bull. Amer. Math. Soc. 61 (1955) 95 | DOI

[24] Wikipedia, Galois connection, electronic reference (2002)

[25] Wikipedia, Book embedding, electronic reference (2006)