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)" [a4], "...a guiding principle to the homotopical foundations of algebraic topology..." [a1], "...a principle or yoga rather than a theorem" [a5], and "...a commonplace of experience among topologists, accepted as obvious" [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 $C$ which can be expressed purely category-theoretically admits a formal dual in the opposite or dual category $C ^ { * }$, which can then be re-interpreted as a notion in the original category $C$; this latter notion is the (Eckmann–Hilton) dual of the original notion. As examples, the notions of monomorphism in $C$ and epimorphism in $C$ are dual, as are the notions of product of objects in $C$ and co-product of objects in $C$. Pursuing the second example, an object $X$ in $C$ (assumed to have zero-mappings) is group-like if there is a morphism $X \times X \rightarrow X$ in $C$, $X \times X$ denoting the product of $X$ with itself, satisfying the group axioms, expressed arrow-theoretically; dually, $X$ in $C$ is co-group-like if there is a morphism $X \rightarrow X \vee X$ in $C$, $X \vee X$ denoting the co-product of $X$ with itself, satisfying the group axioms with arrows reversed. If $X$ is group-like (respectively, co-group-like), the morphism set $\operatorname{mor}( W , X )$ (respectively, $\operatorname{mor}( X , W )$) inherits a natural group structure.

In the category ${\cal H}_*$ of pointed CW-complexes (cf. also CW-complex; 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; Loop space). If the requirements of associativity and existence of inverses are dropped from the group axioms, the resulting objects are $H$-spaces (respectively, co-$H$-spaces; cf. also $H$-space; Co-$H$-space). An important generalization of co-$H$-spaces is obtained by considering the notion of Lyusternik–Shnirel'man category (cf. also Category (in the sense of Lyusternik–Shnirel'man)). Originally conceived as a geometric invariant of a space $X$ ($\operatorname { cat } ( X ) = - 1 +$ the minimum cardinality of an open covering of $X$, each of whose members is contractible in $X$), the definition can be recast (G.W. Whitehead, T. Ganea) in ways that are susceptible to dualization (see [a3] for a useful bibliography, including references to sources for $\operatorname{cat}\,( X )$ and various candidates for a dual, $\operatorname{cocat}( X )$). $X$ is a co-$H$-space if and only if $\operatorname{cat} ( X ) \leq 1$.

At the beginning of their work on duality in 1955, B. Eckmann and P. Hilton studied the category $\mathcal{M}$ of modules (over some ring) and developed two dual notions of module homotopy (cf. also Homotopy): injective homotopy, where, for a module $M$, an injective module containing $M$ plays the role of the "cone" over $M$; and projective homotopy, where a projective module mapping onto $M$ plays the role of the "path space" over $M$. These two versions of homotopy in $\mathcal{M}$ are different but their analogues in ${\cal T}_{*}$, the category of pointed CW-complexes and pointed mappings, are identical owing to the adjunction equivalence

\begin{equation*} \operatorname { map }_{ *}( X \bigwedge Z , Y ) \approx \operatorname { map }_{ *} ( X , \operatorname { map } _ { * } ( Z , Y ) ), \end{equation*}

applied with $Z = [ 0,1 ]$; here $\operatorname{map}_{ *} ( ., . )$ is the morphism set in ${\cal T}_{*}$, suitably topologized, $\wedge$ is smash product and $\approx$ is homeomorphism [a1]. In other words, the duality between projective and injective modules in $\mathcal{M}$ becomes an internal duality in the topological context.

The adjunction equivalence $\operatorname { map } _ { * } ( X \wedge S ^ { 1 } , Y ) \approx \operatorname { map } _ { * } ( X , \operatorname { map } _ { * } ( S ^ { 1 } , Y ) )$, with $S ^ { 1 }$ the circle, induces an adjunction equivalence

\begin{equation*} [ \Sigma X , Y ] \cong [ X , \Omega Y ] \end{equation*}

in ${\cal H}_*$; here $\Sigma$, $\Omega$ and $[ \cdot \ , \ \cdot ]$ denote suspension, loop space and morphism set in ${\cal H}_*$. Using iterated suspensions and loop spaces, one introduces

\begin{equation*} \pi _ { n } ( X , Y ) = [ \Sigma ^ { n } X , Y ] \cong [ X , \Omega ^ { n } Y ], \end{equation*}

simultaneously generalizing the cohomology groups of $X$ (when $Y$ is an Eilenberg–MacLane space) and the homotopy groups of $Y$ (when $X$ is the $0$-sphere and $n \geq 1$). (When $X$ is a Moore space of type $( A , 2 )$, with $A$ an Abelian group, one obtains homotopy groups with coefficients.) The $\pi _ { n } ( X , Y )$ may be generalized to $\pi _ { n } ( \alpha , \beta )$, where $\alpha$, $\beta$ are mappings; namely, $\pi _ { n } ( \alpha , \beta )$ is set equal to the homotopy classes of commutative diagrams $g \circ \alpha = \beta \circ f$. Various relative groups are special cases of this general construction and the standard exact sequences of algebraic topology ensue, in dual pairs. Also, the Postnikov decomposition of a path-connected space $Y$ (where the basic building blocks are Eilenberg–MacLane spaces) and the Moore decomposition of a $1$-connected space $X$ (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 $\varphi : X \rightarrow Y$.

Again appealing to the adjunction equivalence in ${\cal T}_{*}$, one observes that the topological notion of fibration (homotopy lifting property) dualizes to the notion of cofibration (homotopy extension property). Similarly, HELP (homotopy extension and lifting property) dualizes to co–HELP; [a1], Thms. 4; 4$\square ^ { \color{blue} * }$. 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 [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 [a1], Thm. B. Thus, the two illustrious Whitehead theorems are Eckmann–Hilton dual, with dual proofs. (Cf. also 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 $H$-spaces and path-connected co-$H$-spaces, respectively. Thus, the path-connected space $Y$ is an $H$-space if and only if the canonical mapping $Y \rightarrow \Omega \Sigma Y$, adjoint to the identity mapping of $\sum Y$, admits a left homotopy inverse and the path-connected space $X$ is a co-$H$-space if and only if the canonical mapping $\Sigma \Omega X \rightarrow X$, adjoint to the identity mapping of $\Omega X$, 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 $S ^ { 1 }$.

Sometimes the strict dual of a result turns out to admit a surprisingly interesting variant. A theorem of Hilton [a2], Ref. H82, asserts that co-product cancellation fails for finite CW-complexes; there exist $1$-connected $2$-cell CW-complexes $V$, $W$ and a sphere $S$ such that $V \vee S \simeq W \vee S$ but $V \ncong W$. This theorem dualizes straightforwardly to an example of the failure of product cancellation; there exist $1$-connected, 2-stage Postnikov systems $Y , Z$ and an Eilenberg–MacLane space $K$ such that $Y \times K \simeq Z \times K$ but $Y \ncong Z$. The much more delicate question of failure of product cancellation for $1$-connected, finite CW-complexes was studied by P. Hilton and J. Roitberg [a2], Ref. H98. One of their examples leads to the existence of a finite CW-complex which is an $H$-space (indeed, a loop space) not homotopy equivalent to any of the "classical" $H$-spaces. This example, along with other examples of A. Zabrodsky [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 $H$-spaces; these would be finite Postnikov spaces which are co-$H$-spaces. Since a non-contractible, $1$-connected, finite Postnikov space cannot be a co-$H$-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 [a7]), it follows that $S ^ { 1 } \vee \ldots \vee S ^ { 1 }$, the co-product of finitely many circles, is the only path-connected finite Postnikov space admitting a co-$H$-space structure. Thus the dual of a spicy piece of homotopy theory can be rather bland.

The space $S ^ { 1 } \vee \ldots \vee S ^ { 1 }$ calls to mind another example, discussed in [a2], and given here. If $Y$ is a path-connected $H$-space of finite homotopical type (all homotopy groups are finitely generated), then there exists an $H$-space $Y ^ { 1 }$ with $H ^ { 1 } ( Y ^ { 1 } ; \mathbf{Z} ) = 0$ and a homotopy equivalence from $Y^1\times S^1\times \dots \times S^1$ to $Y$. Dually, if $X$ is a path-connected co-$H$-space of finite homological type (all homology groups are finitely generated), there exists a $1$-connected co-$H$-space $X ^ { 1 }$ and a mapping from $X$ to $X ^ { 1 } \vee S ^ { 1 } \vee \ldots \vee S ^ { 1 }$ inducing homology isomorphisms. In 1971, Ganea posed the question of whether $X$ is homotopy equivalent to such a co-product. Recently (1997), N. Iwase has announced a negative answer to this question.

References

[a1]	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
[a2]	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
[a3]	P. Hilton, "Duality in homotopy theory: a restrospective essay" J. Pure Appl. Algebra , 19 (1980) pp. 159–169
[a4]	E.H. Spanier, "Review of: B. Eckmann, Homotopie et dualité, Colloque de topologie algébrique, Louvain 1956, Masson, 1957, 41–53" Math. Reviews , 19 (1958) pp. 570c
[a5]	J. Stasheff, "Hilton–Eckmann duality revisited" Contemp. Math. , 37 (1985) pp. 149–152
[a6]	A. Zabrodsky, "Homotopy associativity and finite CW-complexes" Topology , 9 (1970) pp. 121–128
[a7]	Y. Felix, S. Halperin, J.-M. Lemaire, J.-C. Thomas, "Mod $p$ loop space homology" Invent. Math. , 95 (1989) pp. 247–262