Algebraic Shifting and Basic Constructions on Simplicial Complexes
Eran Nevo
The Hebrew University Institute of Mathematics Givat Ram Jerusalem 91904 Israel Givat Ram Jerusalem 91904 Israel
DOI: 10.1007/s10801-005-4626-0
Abstract
We try to understand the behavior of algebraic shifting with respect to some basic constructions on simplicial complexes, such as union, coning, and (more generally) join. In particular, for the disjoint union of simplicial complexes we prove Δ ( K \?\cup L) = Δ (Δ ( K) \?\cup Δ ( L)) (conjectured by Kalai [6]), and for the join we give an example of simplicial complexes K and L for which Δ ( K* L)\neq Δ (Δ ( K)*Δ ( L)) (disproving a conjecture by Kalai [6]), where Δ denotes the (exterior) algebraic shifting operator. We develop a `homological' point of view on algebraic shifting which is used throughout this work.
Pages: 411–433
Keywords: keywords algebraic shifting; simplicial complexes
Full Text: PDF
References
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2. G. Kalai, “A characterization of f -vectors of families of convex sets in Rd , Part 1: Necessity of Eckhoff's conditions,” Israel J. Math. 48 (1984), 175-195.
3. G. Kalai, “Hyperconnectivity of graphs,” Graphs and Combi. 1 (1985), 65-79.
4. G. Kalai, “Symmetric matroids,” J. Comb.Th.B. 50 (1990), 54-64.
5. G. Kalai, “The diameter of graphs of convex polytopes and f -vector theory,” in: “Applied Geometry and Discrete Mathematics” DIMACS Series in Discrete Math. Comp. Sci. 4 (1991), 387-411.
6. G. Kalai, “Algebraic shifting,” Adv. Stud. in Pure Math. 33 (2002), 121-163.
7. J. Munkres, Elements of algebraic topology, Addison-Wesley, Menlo Park, Calif. 1984.
8. K.S. Sarkaria, “Exterior shifting,” Res. Bulletin of the Punjab University 43 (1993), 259-268.