Topology Atlas | Conferences

Second quandle homology from Schur multiplier (the historical perspective)
by
Jozef H.Przytycki
George Washington University
Coauthors: Rhea Palak Bakshi, Dionne Ibarra, Sujoy Mukherjee, Takefumi Nosaka

The historical perspective starts form my observation that the second homologies of an odd abelian group coincides with the second homology of assocuated Takasaki quandle of the group (a*b=2b-a).

We express the second quandle homology of a quasigroup Alexander quandles in terms of the exterior algebra of X. We present a short self-contained proof of its structure and provide some computational examples. The result is as follows: Let X be an Alexander quandle with (1-t) invertible. Then there is an isomorphism

H2Q(X, Z) = X∧Z X (x∧x- tx∧tx) .

We discuss also and example of connected Alexander quandle which is not a quasigroup:

Cosider the countable direct sum of a group Z indexed by positive numbers:

X=⊕i > 0 Z(i)

with 1_i the identity of Z⁽ⁱ⁾
Let f:X→ X be a epimorphism given on the basis by f(1_i)=1_i-1 for i > 1 and f(1₁)=0. Clearly f is not a monomorphism. We have (1-f)(1_i) = 1_i-1_i-1 and observe that (1-f) is invertible with the inverse given by:

(1-f)-1(1i)=1i+1i-1+...+11.

Now observe that if we put t=1-f then (X, *) with a*b=ta +(1-t)b is a quandle which is not a quasigroup, 1-t=f is not invertible, but which is connected (1-t)X=X.

arXiv:1006.0258 [math.GT]

arXiv:1812.04704 [math.GT]

Date received: May 9, 2019

Copyright © 2019 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 # cbpy-12.