From: "Charles H. Giffen" <chg4k@virginia.edu>
Subject: Re: Homotopy associative co-H space
Date: Tue, 07 Mar 2000 12:57:45 -0500
Newsgroups: sci.math
To: Marco de Innocentis <mdeinnocentis@hotmail.com>

Marco de Innocentis wrote:
> 
> I have trouble understanding the definition of a homotopy associative
> co-H space. I know you have a homotopy associative H space when you
> consider maps from X to Y and Y has a map to itself called *, say,
> which behaves like a group operation up to homotopy. The definition
> of of a h.a. co-H space requires a comultiplication *': X --> X x X,
> or *': X --> X \/ X - but I don't see how this operation helps to
> set up the definition. Can anyone help?
> Thanks,
> 
> Marco
> 
> Sent via Deja.com http://www.deja.com/
> Before you buy.

A co-H space is a space  X  in  Top_0  (pointed spaces), with a
map  c: X --> X.X  = one point union of two copies of  X =
(X x 0) union (0 x X), where  0 in X  is the basepoint.  To be
homotopy associative requires that

	  X  -------->  X.X
	  |     c        |
	  |              |
	  | c            | c.id
	  |              |
	  v              v
	 X.X  ------>  X.X.X 
               id.c

be homotopy commutative in  Top_0  (ie  (c.id)c ~ (id.c)c ).

As an example, consider the circle  X = S^1  in the complex
numbers  C  with basepoint  1,  and let 

	c: X ----> X.X

be defined by  c(z) = (z^2,1) if  Im z >= 0,  c(z) = (1,z^2)
if  Im z <= 0  in  X.X .  Then  (X,c)  is a co-H-space.

--Chuck Giffen