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SUMMARY:The Coalgebra of Chains and the Fundamental Group
DTSTART:20200608T171500
DTEND:20200608T181500
DTSTAMP:20260427T234031Z
UID:747a66c4964a66990084331d7622da8d22eb1ca9f17d6b51ac45efd0
CATEGORIES:Conferences - Seminars
DESCRIPTION:Manuel Rivera\, Purdue University\nI will explain the sense i
 n which the natural algebraic structure of the singular chains on a path-c
 onnected space determines its fundamental group. This is a conceptual obse
 rvation which has several important consequences\, one of them being the f
 ollowing extension of a classical theorem of Whitehead: a continuous map b
 etween path-connected pointed spaces is a weak homotopy equivalence if and
  only if the induced map between the differential graded coalgebras of sin
 gular chains is a cobar-quasi-isomorphism (i.e. a quasi-isomorphism after 
 applying the cobar functor). Another consequence is the following statemen
 t over a field F of arbitrary characteristic: two path-connected pointed s
 paces X and Y are connected by a zig-zag of maps inducing isomorphisms on 
 fundamental groups and on homology with coefficients in any local system i
 f and only if the simplicial cocommutative coalgebras of chains FX and FY 
 are cobar-quasi-isomorphic.\nThe main ingredients needed to formulate and 
 prove these results are: 1) an extension of a classical theorem of Adams w
 hich relates the cobar construction to the based loop space of any path-co
 nnected space 2) the symmetry of the diagonal map and its chain approximat
 ions\, and 3) a theorem of P. Goerss relating Bousfield localization and t
 he simplicial coalgebra of chains.
LOCATION:World Wide Web https://epfl.zoom.us/j/94351048760
STATUS:CONFIRMED
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