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Co-H-spaces are the Eckmann-Hilton duals of H-spaces. They are co-H-objects in the category of pointed topological spaces. Thus a co-H-space is a pointed space, ( X , ϕ ) (X, \phi) , together with a map X X (the ϕ : X → X ∨ X \phi: X \to X \vee X wedge sum), such that is homotopic to p i ∘ ϕ p_i \circ \phi , where 1 X 1_X , are the projections p i , i = 1 , 2 p_i, i = 1, 2 . Alternatively, X ∨ X → X X \vee X \to X is a co-H-space if and only if ( X , ϕ ) (X, \phi) is homotopic to j ∘ ϕ j \circ \phi , where Δ \Delta is the inclusion and j : X ∨ X → X × X j: X \vee X \to X \times X is the Δ : X → X × X \Delta: X \to X \times X diagonal map.
The importance of the notion is that
is a co-H-space if and only if for every space X X , Y Y has a binary operation with unit. Further properties of [ X , Y ] [X, Y] are of interest, in particular being (co)associative and having right and left (co)inverses. In this case ϕ \phi is a X X cogroup. The suspension of a topological space is a cogroup.
Every co-H-space is
path-connected, and its fundamental group is free. Reference
Co-H-spaces, chapter 23 of Handbook of Algebraic Topology, Ioan James (ed.).
Revised on July 8, 2013 16:22:05