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Organizers |
Distributive homology: progress in the last two years
by
Jozef H. Przytycki
George Washington University
Let X be a set and Bin(X) the monoid of binary operations on X. We say that a subset S ⊂ Bin(X) is a distributive set if all pairs of elements *1, *2 ∈ S are right distributive, that is, (a*1b)*2c = (a*2c)*1(b*2c) (we allow *1=*2). We define a (one-term) distributive chain complex C(*) as follows: Cn=Z Xn+1 and the boundary operation ∂(*)n: Cn → Cn-1 is given by: ∂(*)n(x0, ..., xn) = (x1, ..., xn) + ∑i=1n(-1)i(x0*xi, ..., xi-1*xi, xi+1, ..., xn). The homology of this chain complex is called a one-term distributive homology of (X, *) (denoted by Hn(*)(X)). For a distributive set (*1, *2, ..., *k), the multi-term distributive homology Hn(a1, ..., ak)(X) is defined as the homology given by a chain complex (Cn, ∂(a1, ..., ak)) where Cn=ZXn+1 and ∂(a1, ..., ak) = ∑i=1k ai∂(*i).
The definition is less then 2 years old (although modeled on rack and quandle homology) but I am glad to report substantial progress due to work of Y.Berman, A.Crans, M.Jablonowski, G.Mezera, K.Putyra, and A.Sikora.
Paper reference: http://front.math.ucdavis.edu/1105.3700, http://front.math.ucdavis.edu/1109.4850, http://front.math.ucdavis.edu/1111.4772, http://at.yorku.ca/c/b/e/k/13.htm
Date received: March 12, 2012
Copyright © 2012 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbek-21.