Homotopy Transition Cocycles

Details Homotopy Transition Cocycles Homotopy Transition Cocycles

2006-09-07

arxiv.org/abs/math/0609220v2

Mathematics

Algebraic Topology

14 pages, 4 figures

Scientific paper

For locally homotopy trivial fibrations, one can define transition functions $$ g\dab : U\da\cap U\db \to H = H(F)$$ where $H$ is the monoid of homotopy equivalences of $F$ to itself but, instead of the cocycle condition, one obtains only that $g\dab g\dbgam$ is homotopic to $g\dagam$ as a map of $U\da\cap U\db\cap U\dgam$ into $H$. Moreover on multiple intersections, higher homotopies arise and are relevant to classifying the fibration. The full theory was worked out by the first author in his 1965 Notre Dame thesis \cite{wirth:diss}. Here we present it using language that has been developed in the interim. We also show how this points a direction `on beyond gerbes'.

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