Topology Atlas | Conferences

Homology theory in which distributivity replaces associativity
by
Jozef H. Przytycki
George Washington University

Homology theory of associative structures like groups and rings has been zealously studied throughout the past starting from the work of Hopf, Eilenberg, and Hochschild, but non-associative structures, like quandles, were neglected till recently.

Let *:X ×X → X be associative (that is X is a semigroup), then two classical homology theories are group homology with

∂(x1, ...xn)=

(x₂, ..., x_n) +∑_i=1^n-1(-1)ⁱ(x₁, ..., x_i-1, x_i*x_i+1, x_i+2, ..., x_n) +(-1)ⁿ(x₁, ...x_n-1) and Hochschild homology with

∂(x0, x1, ...xn) =

∑_i=0^n-1(-1)ⁱ(x₀, ..., x_i-1, x_i*x_i+1, x_i+2, ..., x_n) +(-1)ⁿ(x_n*x₀, x₁, ...x_n-1)

The condition which can often replace associativity is distributivity and my talk will be devoted to homology of distributive structures with an eye on a hypothetical connection to Khovanov homology.

Consider a right self-distributive 2-argument operation *:X ×X → X (i.e. (a*b)*c = (a*c)*(b*c)); such a universal algebra is called a shelf. Then the boundary operation ∂^(*): RXⁿ → RX^n-1 given by

∂(*)(x1, ...xn)=

∑_i=1ⁿ (-1)ⁱ(x₁*x_i, ..., x_i-1*x_i, x_i+1, ..., x_n), leads to a chain complex, C^(*)(X) (R can be any ring but usually we assume R=Z or R=Z[t]). If (X, *) is a rack, that is * is invertible, then the chain complex is acyclic, but in general it may be nontrivial (A.Sikora conjectures that the homology is always free and for a finite X of N elements with Q left action orbits, its nth homology has a rank (Q-1)N^n-1). If we have two right self-distributive operations on X, say *₁ and *₂ which are distributive one with respect to the other then ∂^(*₁)∂^(*₂) = -∂^(*₂)∂^(*₁) and we can define two term boundary operation ∂ = a∂^(*₁)+b∂^(*₂). If *₂ is a trivial operation x*₂y=x then ∂^(*₁) - ∂^(*₂) leads to the classical (Fenn-Rourke-Sanderson) rack homology, and t∂^(*₁) - ∂^(*₂) to classical (Carter-Kamada-Saito) twisted rack homology.

One can approach our distributive homology from the more general perspective:

Let X be a set and let G(X) denote the set of all binary operations *:X×X → X on X. G(X) has a natural monoid (i.e. a semigroup with identity) structure with a composition *_α*_β given by a(*_α*_β)b=(a*_αb)*_βb, and the identity element operation *₀ given by a*₀b=a. We are mostly interested in submonoids of G(X) whose all pairs of elements are right distributive. that is for any pair of elements *_α, and *_β of the monoid, the operation *_β is right distributive with respect to *_α, ( (a*_αb)*_βc = (a*_βc)*_α(b*_βc)). Such a monoid is called a monoid of shelf operations (resp. group of rack operations). The classical example of a shelf submonoid is generated by ∧ and ∨ operations of a distributive lattice on X.

For a given finite number of elements of a shelf submonoid *₁, ..., *_k we define a k-term derivative ∂^{(a₁, ..., a_k)} = a₁∂⁽¹⁾ + ... +a_k∂^(k) and study related homology and interrelations between various homologies.

If our operations satisfy idempotency condition x*x=x then (after Carter-Kamada-Saito) one can define k-term degenerate and quandle homology.

In the talk I will explore the above ideas and illustrate by two recent theorems about classical quandle homology (here G is an abelian group and the Takasaki quandle, T(G)=(G, *) is defined by a*b=2b-a.

Theorem 1. tor H_n^Q(T(Z_p)) = Z_p^f_n where f_n is delayed Fibonacci sequence f_n = f_n-1 + f_n-3,  and f(1)=f(2)=0, f(3)=1.
Theorem 2. H_n^Q(T(G) = G ∧G where ∧ is the exterior product, and G has an odd order.

Finally back to associativity; can one bring some of the above ideas back to classical objects? Yes, one can try out-associativity, that is (a*₁b)*₂c = a*₂(b*₁c) and linear combinations of various boundary operators.

Date received: November 28, 2010

Copyright © 2010 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 # cbbr-24.