Shifting chain maps in quandle homology and cocycle invariants

Quandle homology theory has been developed and cocycles have used to define invariants of oriented classical or surface links. We introduce a shifting chain map σ \sigma on each quandle complex that lowers the dimensions by one. By using its pull-back alttext="sigma Superscript normal ♯"> ♯<!-- ♯ </mml:msup> encoding="application/x-tex">\sigma ^\sharp , alttext="2"> 2 encoding="application/x-tex">2 -cocycle alttext="phi"> ϕ<!-- ϕ encoding="application/x-tex">\phi gives us alttext="3"> 3 encoding="application/x-tex">3 Baseline phi"> ^\sharp \phi . For links in -space, we explore relation between their associated with shadow alttext="4"> 4 encoding="application/x-tex">4 how powerful are. Algebraic behavior shifting maps for low-dimensional (co)homology groups is also discussed.