Timeline for When can you desuspend a homotopy cogroup?

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Nov 5, 2009 at 16:44	comment	added	Charles Rezk		Oh, I hadn't even though about the "cogrouplike" condition. I guess cogrouplike should mean (i,c): Y v Y --> Y v Y is a weak equivalence, where on the first factor this is inclusion into one of the summands, and on the other one is the comultiplication (this is just dual to the "grouplike" condition on an H-space). If Y is simply connected, we just have to check that the map induces iso in homology; but this is necessarily so, since co-A-infinity structure makes H_*Y into a comonoid in abelian groups, which automatically has to be a cogroup.
Nov 5, 2009 at 10:27	comment	added	Oscar Randal-Williams		Presumably in general there should be a map of co-A-infinity spaces ΣX -> Y which is an equivalence when Y is "cogrouplike". Any ideas as to what "cogrouplike" might be? From the remark above it seems that 2-connected implies cogrouplike, so it ought to be a condition on the 2-skeleton.
Nov 5, 2009 at 1:45	history	edited	Charles Rezk	CC BY-SA 2.5	Reference to commentary.
Nov 5, 2009 at 1:33	history	edited	Charles Rezk	CC BY-SA 2.5	More clarification.
Nov 5, 2009 at 1:26	history	answered	Charles Rezk	CC BY-SA 2.5