1. CJM 2011 (vol 63 pp. 1345)
This paper gives a characterization of homotopy fibres of inverse image maps on groupoids of torsors that are induced by geometric morphisms, in terms of both pointed torsors and pointed cocycles, suitably defined. Cocycle techniques are used to give a complete description of such fibres, when the underlying geometric morphism is the canonical stalk on the classifying topos of a profinite group $G$. If the torsors in question are defined with respect to a constant group $H$, then the path components of the fibre can be identified with the set of continuous maps from the profinite group $G$ to the group $H$. More generally, when $H$ is not constant, this set of path components is the set of continuous maps from a pro-object in sheaves of groupoids to $H$, which pro-object can be viewed as a ``Grothendieck fundamental groupoid".
Keywords:pointed torsors, pointed cocycles, homotopy fibres
Categories:18G50, 14F35, 55B30
2. CJM 2011 (vol 64 pp. 102)
|Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies|
We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.
Keywords:quandle cocycle invariant, knotted handlebody, spatial graph
Categories:57M27, 57M15, 57M25