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Progress in one term distributive homology
by
Jozef H. Przytycki
George Washington University
Coauthors: Alissa S.Crans, Krzysztof K.Putyra

I will discuss the progress made in the study of one term distributive homology made in the last few months. We computed completely the one term distributive homology for a spindle of 2 block decomposition of the type X=X₀\sqcup b. An important tool is the simplified version of the Künneth formula for the degenerate part of rack homology of spindles (see K.Putyra talk). We also show that one term distributive homology of a finite spindle can have any finite torsion. The first example in which a torsion contains Z₂⊕Z₄, the motivation for the general, multi-block construction, is given below:

Consider a 8-element spindle (X;*) with operation * given by the following table (notice blocks of size 5 and 3):

æ ç ç ç ç ç ç ç ç ç è 1 2 3 4 5 7 6 6 1 2 3 4 5 7 6 6 1 2 3 4 5 7 6 6 1 2 3 4 5 7 6 6 1 2 3 4 5 7 6 6 2 3 4 1 1 6 7 8 2 3 4 1 1 6 7 8 2 3 4 1 1 6 7 8 ö ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø .

Then H₀^(*)(X)=Z² and

H1(X) = Z2 ⊕Z27 ⊕Z4

H2(X) = Z16 ⊕Z248 ⊕Z48

H3(X) = Z128 ⊕Z2392 ⊕Z464

Similarly consider a 17-element spindle (X;*) with operation * given by the following table (notice blocks of size 3, 5 and 9), to get H₁(X) = Z²⁰ ⊕Z₂ ⊕Z₄ ⊕Z₈.

Paper reference: http://front.math.ucdavis.edu/1109.4850, http://arxiv.org/abs/1111.4772, http://front.math.ucdavis.edu/1105.3700

Date received: May 1, 2013

Copyright © 2013 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 # cbhe-15.