Topology Atlas | Conferences
Knots in Washington XXXII, Categorification of Knots, Algebras, and Quandles; Quantum Computing
April 29 - May 1, 2011
George Washington University
Washington, DC, USA |
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Organizers
Valentina Harizanov (GWU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU, NSF), Radmila Sazdanovic (U.Penn), Alexander Shumakovitch (GWU), Hao Wu (GWU)
Conference Homepage |
Weak and very weak simplicial modules: application to homology of distributive structures
by
Jozef H. Przytycki
George Washington University
In classical homological algebra a simplicial module (Mn, di, si) is a collection of
R-modules Mn, n ≥ 0, together with face maps di:Mn→ Mn-1 and degenerate maps
si: Mn→ Mn+1, 0 ≤ i ≤ n, which satisfy the following properties:
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(1) didj = dj-1di for i < j. |
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(2) sisj=sj+1si, 0 ≤ i ≤ j ≤ n, |
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(3) disj = sj-1di if i < j and sjdi-1 if i > j+1 |
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(Mn, di) satisfying (1) is called a presimplicial module and leads to the chain complex
(Mn, ∂n) with ∂n = ∑i=0n(-1)idi.
If (X, *) is a right selfdistributive magma, called by Alissa Crans a shelf
(i.e. (a*b)*c=(a*c)*(b*c)) then we define (motivated
by Fenn, Rourke and Sanderson, and Carter, Kamada, and Saito), a presimplicial module
(Mn, di), where Mn=RXn+1, d0(x0, x1, ..., xn) = (x1, ..., xn) and for i > 0
di(x0, x1, ..., xn)=(x0*xi, ..., xi-1*xi, xi+1, ..., xn).
(We shift by 1 chain modules of FRS-CKS; that is Cn+1(X)=Mn(X)=RXn+1, to conform
to the classical definition of a presimplicial module).
The homology Hn(*)(X) of
the chain complex (Mn, ∂n) are called one term distributive homology and their
computation is a topic of a joint paper with Adam Sikora.
We can define degeneracy maps si: Mn→ Mn+1 by si(x0, ..., xn)=(x0, ..., xi-1, xi, xi, xi+1, ..., xn),
and notice that (Mn, di, si) satisfies axioms (1)-(3) of simplicial set. We can still consider
the submodules MDn ⊂ Mn defined by MDn=span(s0Mn-1, ..., sn-1Mn-1). We check
that (MDn, ∂n) is a subchain complex of (Mn, ∂n) iff
(4') is equivalent here with idempotency condition a*a=a for any a ∈ X (a shelf with idempotency
condition is called a spindle).
In general (Mn, di, si) satisfying (1)-(3) and (4') will be called a weak simplicial module.
Notice that in a weak simplicial module we have boundary preserving filtration:
1 ⊂ F0 ⊂ ... ⊂ Fn-2 ⊂ Fn-1=MnD, with
Fp=FpMn = span(s0Mn-1, ..., spMn-1).
This observation allows us to compute one term distributive homology for (X, *L), where *L is
the left trivial operation a*Lb=b. We check that Hn(*L)(X) is a free R-module with
a basis (x0, x1, ..., xn)-(x1, x1, ..., xn), where x0 ≠ x1, in particular, if X is a finite set,
Hn(*L)(X) = R(|X|-1)|X|n. To demonstrate this we just observe that F0Mn is acyclic and
that ∂ sends Mn/F0Mn to zero.
In the case of spindles the homology splits:
Hn(*)(X) = HnD(X) ⊕HQn(X) where HQn(X)=Hn(Mn/MnD).
We ask whether the split holds generally for a weak simplicial module (recall that for a simplicial module,
HDn(X)=0.). Assume now that only conditions (1)-(3) hold. In such a case we call (Mn, di, si)
a very weak (frail) simplicial module. MnD is not necessary a subchain complex of (Mn, ∂n).
To remedy this we consider level maps ti: Mn → Mn defined by ti=disi - di+1si and define
boundary preserving filtration 1 ⊂ Ft0 ⊂ ... ⊂ Ftn-2 ⊂ Ftn-1=Mtn, and
1 ⊂ FtD0 ⊂ ... ⊂ FtDn-2 ⊂ FtDn-1=MtDn, by
Fpt=FptMn = span(t0Mn, ..., tpMn), 0 ≤ i ≤ n, and
FtDp = span(Ftp, s0Mn-1, ..., spMn-1) for p < n,
FtDn = (FtDn-1, tnMn).
We propose to call (MntD, ∂n) a generalized degenerate subchain complex in the
theory of very weak simplicial modules and use it in calculations of homology of
shelfs and racks. For example, we use it (or precisely F0t ⊂ FtD0 ⊂ Mn),
to compute one term homology Hn*g(X) where a*gb=g(b) and g2=g. For a more general case
of retraction of shelf r:X→ A, we compute one term homology in a paper with A.Sikora.
Paper reference: http://at.yorku.ca/c/b/b/r/24.htm
Date received: April 19, 2011
Copyright © 2011 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 # cbcc-17.