Subject: cup_i-products with local coefficients? From: "Ronald Brown" Date: Mon, 27 Sep 2004 12:39:26 +0100 To: "Don Davis" reply to r.brown@bangor.ac.uk The following is suggested by Subject: response to Karoubi's question From: "FERNANDO MURO JIMENEZ" Date: Fri, 24 Sep 2004 19:23:53 +0200 Here is a project for someone, who is willing to learn about crossed complexes! A series of papers by Philip Higgins and me sets up many basic properties of the category of crossed complexes (a notion going back to Blakers, JHC Whitehead, Huebschmann, Lue, ...) including tensor products and homotopies, and classifying space. This category also gives a setting for a generalisation of cohomology with local coefficients (see the paper on the classifying space ``The classifying space of a crossed complex'', {\em Math. Proc. Camb. Phil. Soc.} 110 (1991) 95-120. ), since crossed complexes can, unlike the traditional chain complexes with group of operators, include full information on a presentation of a group or groupoid. An exposition of acyclic models for crossed complexes has also been written up recently: it is hinted at in the above paper. Formulae for the Eilenberg-Zilber theorem in this category have been given by Andy Tonks (thesis, available from Bangor math web site (/research/theses) , and JPAA). So if \Pi K is the fundamental crossed complex of a simplicial set K, there is a diagonal map AW: \Pi K \to \Pi K \otimes \Pi K. This of course is not cocommutative, but there is, by acyclic models, a crossed complex homotopy h: T \circ AW \simeq AW. Can this process be pursued further as in Steenrod's original approach? Is there a serious obstruction to so doing? In any case it is possible (probable?) that the homotopy h above contains significant information. For references on crossed complexes, see http://www.bangor.ac.uk/~mas010/hdaweb2.htm Ronnie Brown