By Christos A. Athanasiadis, Victor V. Batyrev, Dimitrios I. Dais, Martin Henk, and Francisco Santos

This quantity includes unique learn and survey articles stemming from the Euroconference "Algebraic and Geometric Combinatorics". The papers speak about a variety of difficulties that illustrate interactions of combinatorics with different branches of arithmetic, comparable to commutative algebra, algebraic geometry, convex and discrete geometry, enumerative geometry, and topology of complexes and in part ordered units. one of the subject matters lined are combinatorics of polytopes, lattice polytopes, triangulations and subdivisions, Cohen-Macaulay phone complexes, monomial beliefs, geometry of toric surfaces, groupoids in combinatorics, Kazhdan-Lusztig combinatorics, and graph shades. This e-book is geared toward researchers and graduate scholars attracted to a variety of features of contemporary combinatorial theories

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D1/; = (1/;J, 1/;j , 1/;5, 1/;? , ... ,1/;'0, 1/;1) is called the boundary of 1/;. In low dimensions n-cubes, face and degeneracy operators can be illustrated by the following figures. • a O-cube (Ja cP a. 9' .. a degenerate I-cube CPo ••--')-~. CPl I-cube 1/;? ~{~J~l 1/;5 (1/;5)1 = (1/;Do 2-cube ¢[8}¢ CPl CPo (JCPl (Jcpo CPl CPo cP ¢OhJ~rl (N (Jcpo CPo (JCPl cP CPl degenerate 2-cubes 21 a (Ja a (Ja (f(Ja = a(Ja (Ja a (Ja a -+---1f 15 3-cube Let Q, Q' be cubical sets. A morphism i : Q --+ Q' is a sequence of maps, in : Qn --+ Q~ , commuting with face and degeneracy operators .

Applications. 7). 7). A morphism morphism ii :: A A ---+ +X X of of C C isis aa strong strong there is is aa morphism morphism rr :: X X ---+ 4 A such such that that deformation retract retract if there deformation A A r i == IdA I d A and and ir ir ~ c Id I d xx , ri where ir i r and and II dxx are are viewed viewed as as morphisms morphisms under under A, A, ir, ir, I d xx :: ii ---t 4 where i2.. 8). 8). A A morphism morphism isis aa trivial trivial cofibration cofibration ifif it it is is aa Definition cofibration cofibration and and aa homotopy homotopy equivalence.

We obtain a commutative diagram ho Ao h ~ A' hi ;. A2 Bo / . B' g2 ~ . B2 h2 where k , l are cofibrations and u, v are homotopy equivalences. Since by assumption h2 is a homotopy equivalence, so is hi. 5) ~ A' ;. a i A" if ~ A2 - - - - - - A i~ where the upper 'square' is chosen to be a pushout and u ' is induced by the pushout property. 5) is a pushout. 4). Similarly, v' is a homotopy equivalence. Using the pushout property we obtain an induced map h" : A" ----t B". Now we are in a situation where the special case of the Gluing Theorem applies.

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