Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius

Andrea Brini

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-A, Issue: 1, page 143-172
  • ISSN: 0392-4033

Abstract

top
Si introduce la nozione combinatoria di connessione di Galois tra insiemi parzialmente ordinati e se ne descrivono i principali risultati di caratterizzazione; questi risultati aprono la strada alla comprensione del profondo legame che sussiste tra la nozione connessione di Galois ed il Criterio di Omotopia di Quillen. Si introduce quindi la nozione di funzione di Möbius di un reticolo finito L e se ne discute brevemente, anche tramite un esempio significativo, la cruciale importanza nell’ambito della Combinatoria Enumerativa e della Probabilità Discreta. Dopo aver riconosciuto che i valori della funzioni di Möbius possono essere interpretati come «Caratteristiche di Eulero» di opportuni complessi, a titolo di esempio e di applicazione di metodi topologici alla combinatoria degli insiemi parzialmente ordinati, si presentano e si dimostrano le versioni topologiche di due classici Teoremi: il «Teorema del Cross-Cut» ed il «Teorema di annullamento per reticoli non fortemente complementati».

How to cite

top

Brini, Andrea. "Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius." Bollettino dell'Unione Matematica Italiana 7-A.1 (2004): 143-172. <http://eudml.org/doc/289415>.

@article{Brini2004,
abstract = {The notion of Galois Connections between partially ordered sets is introduced, together with a presentation of some of its main characterizations. This leads to a true understanding of the deep connection that links Galois Connections to Quillen’s Homotopy Type Equivalence Theorem. Furthermore, the notion of Möbius functions of finite lattices is discussed, in order to show its crucial role in Enumerative Combinatorics over Finite Posets and Discrete Probability Theory. Since the values of the Möbius function of a finite lattice may be regarded as reduced Euler Characteristic of suitable topological spaces, a wide variety of combinatorial results can be proved by topological methods. We exploit this point of view by providing elementary proofs of two classical theorems: the «Cross-Cut Theorem» of Rota and the «Vanishing Theorem for not-strongly complemented lattices» of Crapo.},
author = {Brini, Andrea},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {ita},
month = {4},
number = {1},
pages = {143-172},
publisher = {Unione Mastematica Italiana},
title = {Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius},
url = {http://eudml.org/doc/289415},
volume = {7-A},
year = {2004},
}

TY - JOUR
AU - Brini, Andrea
TI - Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/4//
PB - Unione Mastematica Italiana
VL - 7-A
IS - 1
SP - 143
EP - 172
AB - The notion of Galois Connections between partially ordered sets is introduced, together with a presentation of some of its main characterizations. This leads to a true understanding of the deep connection that links Galois Connections to Quillen’s Homotopy Type Equivalence Theorem. Furthermore, the notion of Möbius functions of finite lattices is discussed, in order to show its crucial role in Enumerative Combinatorics over Finite Posets and Discrete Probability Theory. Since the values of the Möbius function of a finite lattice may be regarded as reduced Euler Characteristic of suitable topological spaces, a wide variety of combinatorial results can be proved by topological methods. We exploit this point of view by providing elementary proofs of two classical theorems: the «Cross-Cut Theorem» of Rota and the «Vanishing Theorem for not-strongly complemented lattices» of Crapo.
LA - ita
UR - http://eudml.org/doc/289415
ER -

References

top
  1. AIGNER, M., Combinatorial Theory, Springer-Verlag, 1978. 
  2. BARNABEI, M. - BRINI, A. - ROTA, G.-C., Theory of Möbius functions (in russo), Uspehi Mat. Nauk, 41(1986), 113-157. English translation: Theory of Möbius functions, Russian Math. Translations (London Math. Soc.), 41 (1987), 113-157. MR542445
  3. BENDER, E. A.- GOLDMAN, J. R., On application of Möbius inversion in combinatorial analysis, Amer. Math. Monthly, 82(1975), 789-803. MR854241
  4. BJORNER, A., Homotopy types of posets and lattice complementation, J. Combin. Theory (A), 30(1981), 90-100. Zbl0442.55011MR376360
  5. BJORNER, A., Topological Methods, in «Handbook of Combinatorics, vol. II» (R. L. Graham, M. Grotschel, L. Lovasz, Eds.), pp. 1821-1872, North-Holland, Amsterdam, 1995. MR607041
  6. BRINI, A., Some Homological Properties of Partially Ordered Sets, Advances in Math., 43(1982), 197-201. Zbl0484.06003MR1373690
  7. CERASOLI, M. - EUGENI, F. - PROTASI, M., «Elementi di Matematica Discreta», Zanichelli, 1988. MR644672
  8. CRAPO, H., The Möbius function of a lattice, J. Combin. Theory (A), 1(1966), 126-131. Zbl0146.01601MR1093460
  9. FOLKMAN, J., The homology groups of a lattice, J. Math. Mech., 15(1966), 631-636. Zbl0146.01602MR193018
  10. LAKSER, H., The homology of a lattice, Disc. Math., 1(1971), 187-192. Zbl0227.06002MR188116
  11. MATHER, J., Invariance of the homology of a lattice, Proc. Amer. Math. Soc., 17(1966), 1120-1124. Zbl0147.42102MR288755
  12. MAUNDER, C. R. F., «Algebraic Topology», Van Nostrand, 1970. MR202645
  13. MUNKRES, J. R., «Elements of Algebraic Topology», Addison-Wesley, 1984. MR1402473
  14. ORE, O., Galois connections, Trans. Amer. Math. Soc., 55 (1944), 493- 513. Zbl0060.06204
  15. ROTA, G.-C., On the foundations of combinatorial theory I. Theory of Möbius functions, Z. Wahrsch. Verw. Gebiete, 2(1964), 340-368. MR10555
  16. QUILLEN, D., Higher algebraic K-theory, I, in «Algebraic K-Theory, 1», Lecture Notes in Mathematics, No. 341, pp. 85-147, Springer-Verlag, Berlin, 1973. MR174487
  17. QUILLEN, D., Homotopy properties of posets of non-trivial p-subgroups of a group, Advances in Math., 28(1978), 101-128. Zbl0388.55007MR338129
  18. STANLEY, R. P., «Enumerative Combinatorics. Volume I», Wadsworth and Brooks, 1986. MR493916
  19. WALKER, J. W., Homotopy Type and Euler Characteristic of Partially Ordered sets, Europ. J. Combinatorics, 2(1981), 373-384. Zbl0472.06004MR847717

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.