View source for Eckmann-Hilton duality

A duality principle variously described as  "...a metamathematical principle that corresponding to a theorem there is a dual theorem (each of these dual theorems being proved separately)"  [[#References|[a4]]],  "...a guiding principle to the homotopical foundations of algebraic topology..."  [[#References|[a1]]],  "...a principle or yoga rather than a theorem"  [[#References|[a5]]], and  "...a commonplace of experience among topologists, accepted as obvious"  [[#References|[a3]]]. The duality provides a categorical point of view for clarifying and unifying various aspects of pointed homotopy theory, but is often heuristic rather than strictly categorical.

Any notion (definition, theorem, etc.) in a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200201.png" /> which can be expressed purely category-theoretically admits a formal dual in the opposite or [[dual category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200202.png" />, which can then be re-interpreted as a notion in the original category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200203.png" />; this latter notion is the (Eckmann–Hilton) dual of the original notion. As examples, the notions of monomorphism in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200204.png" /> and epimorphism in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200205.png" /> are dual, as are the notions of product of objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200206.png" /> and co-product of objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200207.png" />. Pursuing the second example, an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200208.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e1200209.png" /> (assumed to have zero-mappings) is group-like if there is a [[Morphism|morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002012.png" /> denoting the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002013.png" /> with itself, satisfying the group axioms, expressed arrow-theoretically; dually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002015.png" /> is co-group-like if there is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002018.png" /> denoting the co-product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002019.png" /> with itself, satisfying the group axioms with arrows reversed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002020.png" /> is group-like (respectively, co-group-like), the morphism set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002021.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002022.png" />) inherits a natural [[Group|group]] structure.

In the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002023.png" /> of pointed CW-complexes (cf. also [[CW-complex|CW-complex]]; [[Pointed space|Pointed space]]) and pointed homotopy classes of mappings,  "product"  is Cartesian product and  "co-product"  is one-point union. The most familiar group-like (respectively, co-group-like) objects are loop spaces (respectively, suspensions) (cf. also [[Suspension|Suspension]]; [[Loop space|Loop space]]). If the requirements of associativity and existence of inverses are dropped from the group axioms, the resulting objects are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002024.png" />-spaces (respectively, co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002025.png" />-spaces; cf. also [[H-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002026.png" />-space]]; [[Co-H-space|Co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002027.png" />-space]]). An important generalization of co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002028.png" />-spaces is obtained by considering the notion of Lyusternik–Shnirel'man category (cf. also [[Category (in the sense of Lyusternik–Shnirel'man)|Category (in the sense of Lyusternik–Shnirel'man)]]). Originally conceived as a geometric invariant of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002030.png" /> the minimum cardinality of an open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002031.png" />, each of whose members is contractible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002032.png" />), the definition can be recast (G.W. Whitehead, T. Ganea) in ways that are susceptible to dualization (see [[#References|[a3]]] for a useful bibliography, including references to sources for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002033.png" /> and various candidates for a dual, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002034.png" />). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002035.png" /> is a co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002036.png" />-space if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002037.png" />.

At the beginning of their work on duality in 1955, B. Eckmann and P. Hilton studied the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002038.png" /> of modules (over some ring) and developed two dual notions of module homotopy (cf. also [[Homotopy|Homotopy]]): injective homotopy, where, for a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002039.png" />, an injective module containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002040.png" /> plays the role of the  "cone"  over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002041.png" />; and projective homotopy, where a projective module mapping onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002042.png" /> plays the role of the  "path space"  over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002043.png" />. These two versions of homotopy in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002044.png" /> are different but their analogues in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002045.png" />, the category of pointed CW-complexes and pointed mappings, are identical owing to the adjunction equivalence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002046.png" /></td> </tr></table>

applied with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002047.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002048.png" /> is the morphism set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002049.png" />, suitably topologized, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002050.png" /> is smash product and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002051.png" /> is homeomorphism [[#References|[a1]]]. In other words, the duality between projective and injective modules in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002052.png" /> becomes an internal duality in the topological context.

The adjunction equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002053.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002054.png" /> the circle, induces an adjunction equivalence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002055.png" /></td> </tr></table>

in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002056.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002059.png" /> denote suspension, loop space and morphism set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002060.png" />. Using iterated suspensions and loop spaces, one introduces

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002061.png" /></td> </tr></table>

simultaneously generalizing the cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002062.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002063.png" /> is an [[Eilenberg–MacLane space|Eilenberg–MacLane space]]) and the homotopy groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002064.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002065.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002066.png" />-sphere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002067.png" />). (When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002068.png" /> is a [[Moore space|Moore space]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002069.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002070.png" /> an Abelian group, one obtains homotopy groups with coefficients.) The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002071.png" /> may be generalized to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002072.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002074.png" /> are mappings; namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002075.png" /> is set equal to the homotopy classes of commutative diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002076.png" />. Various relative groups are special cases of this general construction and the standard exact sequences of [[Algebraic topology|algebraic topology]] ensue, in dual pairs. Also, the Postnikov decomposition of a path-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002077.png" /> (where the basic building blocks are Eilenberg–MacLane spaces) and the Moore decomposition of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002078.png" />-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002079.png" /> (where the basic building blocks are Moore spaces) are dual to one another; these two decompositions appear as special cases of what Eckmann and Hilton describe as the homotopy decomposition, respectively the homology decomposition of a map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002080.png" />.

Again appealing to the adjunction equivalence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002081.png" />, one observes that the topological notion of [[Fibration|fibration]] (homotopy lifting property) dualizes to the notion of [[Cofibration|cofibration]] (homotopy extension property). Similarly, HELP (homotopy extension and lifting property) dualizes to co–HELP; [[#References|[a1]]], Thms. 4; 4<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002082.png" />. But HELP leads to the theorem of J.H.C. Whitehead that a mapping of path-connected CW-complexes inducing isomorphisms on homotopy groups is a homotopy equivalence [[#References|[a1]]], Thm. A, while co–HELP leads to another Whitehead theorem that a mapping of path-connected nilpotent CW-complexes inducing isomorphisms on homology groups is a homotopy equivalence [[#References|[a1]]], Thm. B. Thus, the two illustrious Whitehead theorems are Eckmann–Hilton dual, with dual proofs. (Cf. also [[Homotopy group|Homotopy group]].)

It is not true, however, that dual theorems necessarily admit dual proofs. An example is afforded by theorems of I.M. James and T. Ganea characterizing path-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002085.png" />-spaces and path-connected co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002086.png" />-spaces, respectively. Thus, the path-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002087.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002088.png" />-space if and only if the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002089.png" />, adjoint to the identity mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002090.png" />, admits a left homotopy inverse and the path-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002091.png" /> is a co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002092.png" />-space if and only if the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002093.png" />, adjoint to the identity mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002094.png" />, admits a right homotopy inverse. No known proof of either theorem dualizes to a proof of the other.

It is also possible that the dual of a theorem is false. As an example, consider the well-known result that the suspension of the loop space of a sphere is homotopy equivalent to a co-product of spheres. The dual would assert that the loop space of the suspension of an Eilenberg–MacLane space is homotopy equivalent to a product of Eilenberg–MacLane spaces. However, this assertion fails already in the case that the Eilenberg–MacLane space is the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002095.png" />.

Sometimes the strict dual of a result turns out to admit a surprisingly interesting variant. A theorem of Hilton [[#References|[a2]]], Ref. H82, asserts that co-product cancellation fails for finite CW-complexes; there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002096.png" />-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002097.png" />-cell CW-complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e12002099.png" /> and a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020100.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020101.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020102.png" />. This theorem dualizes straightforwardly to an example of the failure of product cancellation; there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020103.png" />-connected, 2-stage Postnikov systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020104.png" /> and an Eilenberg–MacLane space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020106.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020107.png" />. The much more delicate question of failure of product cancellation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020108.png" />-connected, finite CW-complexes was studied by P. Hilton and J. Roitberg [[#References|[a2]]], Ref. H98. One of their examples leads to the existence of a finite CW-complex which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020110.png" />-space (indeed, a loop space) not homotopy equivalent to any of the  "classical"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020111.png" />-spaces. This example, along with other examples of A. Zabrodsky [[#References|[a6]]] helped usher in a new subdiscipline of homotopy theory, that of  "finite H-spaces"  and  "finite loop spaces" .

A conscientious Eckmann–Hilton dualist might enquire about the existence of duals of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020112.png" />-spaces; these would be finite Postnikov spaces which are co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020113.png" />-spaces. Since a non-contractible, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020114.png" />-connected, finite Postnikov space cannot be a co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020115.png" />-space (the Lyusternik–Shnirel'man category of such a space is infinite according to Y. Félix, S. Halperin, J.-M. Lemaire and J.-C. Thomas [[#References|[a7]]]), it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020116.png" />, the co-product of finitely many circles, is the only path-connected finite Postnikov space admitting a co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020117.png" />-space structure. Thus the dual of a spicy piece of homotopy theory can be rather bland.

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020118.png" /> calls to mind another example, discussed in [[#References|[a2]]], and given here. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020119.png" /> is a path-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020120.png" />-space of finite homotopical type (all homotopy groups are finitely generated), then there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020121.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020122.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020123.png" /> and a homotopy equivalence from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020124.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020125.png" />. Dually, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020126.png" /> is a path-connected co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020127.png" />-space of finite homological type (all homology groups are finitely generated), there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020128.png" />-connected co-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020129.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020130.png" /> and a mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020131.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020132.png" /> inducing homology isomorphisms. In 1971, Ganea posed the question of whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020133.png" /> is homotopy equivalent to such a co-product. Recently (1997), N. Iwase has announced a negative answer to this question.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.P. May,   "The dual Whitehead theorems"  I.M. James (ed.) , ''Topological Topics. Articles on Algebra and Topology presented to Prof. P.J. Hilton in celebration of his sixtieth birthday'' , ''London Math. Soc. Lecture Notes'' , '''86''' , Cambridge Univ. Press  (1983)  pp. 46–54</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Mislin,   "Essay on Hilton's work in topology"  I.M. James (ed.) , ''Topological Topics. Articles on Algebra and Topology presented to Prof. P.J. Hilton in celebration of his sixtieth birthday'' , ''London Math. Soc. Lecture Notes'' , '''86''' , Cambridge Univ. Press  (1983)  pp. 15–30</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Hilton,   "Duality in homotopy theory: a restrospective essay"  ''J. Pure Appl. Algebra'' , '''19'''  (1980)  pp. 159–169</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.H. Spanier,   "Review of: B. Eckmann, Homotopie et dualité, Colloque de topologie algebrique, Louvain 1956, Masson, 1957, 41–53"  ''Math. Reviews'' , '''19'''  (1958)  pp. 570c</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Stasheff,   "Hilton–Eckmann duality revisited"  ''Contemp. Math.'' , '''37'''  (1985)  pp. 149–152</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Zabrodsky,   "Homotopy associativity and finite CW-complexes"  ''Topology'' , '''9'''  (1970)  pp. 121–128</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Y. Felix,   S. Halperin,   J.-M. Lemaire,   J.-C. Thomas,   "Mod <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020134.png" /> loop space homology"  ''Invent. Math.'' , '''95'''  (1989)  pp. 247–262</TD></TR></table>