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
which can be expressed purely category-theoretically admits a
formal dual in the opposite or dual category
,
which can then be re-interpreted as a notion in the original category
;
this latter notion is the (Eckmann–Hilton)
dual
of the
original notion. As examples, the notions of monomorphism in
and epimorphism in
are dual, as are the notions of product of objects in
and co-product of objects in
.
Pursuing the second example, an object
in
(assumed to have zero-mappings) is
group-like
if there is a
morphism
in
,
denoting the product of
with itself, satisfying the group axioms, expressed
arrow-theoretically; dually,
in
is
co-group-like
if there is a morphism
in
,
denoting the co-product of
with itself, satisfying the group axioms with arrows reversed. If
is group-like (respectively, co-group-like), the morphism set
(respectively,
)
inherits a natural
group
structure.
In the category
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
-spaces
(respectively,
co-
-spaces;
cf. also
-space;
Co-
-space).
An important generalization of
co-
-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
(
the minimum cardinality of an open covering of
,
each of whose members is contractible in
),
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
and various candidates for a dual,
).
is a
co-
-space
if and only if
.
At the beginning of their work on duality in
1955,
B. Eckmann
and
P. Hilton
studied the category
of modules (over some ring) and developed two dual notions of
module homotopy
(cf. also
Homotopy):
injective homotopy,
where, for a module
,
an injective module containing
plays the role of the
"cone"
over
;
and
projective homotopy,
where a projective module mapping onto
plays the role of the
"path space"
over
.
These two versions of homotopy in
are different but their analogues in
,
the category of pointed
CW-complexes
and pointed mappings, are identical owing to the
adjunction equivalence
applied with
;
here
is the morphism set in
,
suitably topologized,
is
smash product
and
is homeomorphism
[a1].
In other words, the duality between projective and injective modules
in
becomes an
internal duality
in the topological context.
The adjunction equivalence
,
with
the circle, induces an adjunction equivalence
in
;
here
,
and
denote suspension, loop space and morphism set in
.
Using iterated suspensions and loop spaces, one introduces
simultaneously generalizing the cohomology groups of
(when
is an
Eilenberg–MacLane space)
and the homotopy groups of
(when
is the
-sphere
and
).
(When
is a
Moore space
of type
,
with
an Abelian group, one obtains homotopy groups with coefficients.) The
may be generalized to
,
where
,
are mappings; namely,
is set equal to the
homotopy classes of commutative diagrams
.
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
(where the basic building blocks are Eilenberg–MacLane
spaces) and the
Moore decomposition
of a
-connected
space
(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
.
Again appealing to the adjunction equivalence in
,
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
.
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
-spaces and path-connected co-
-spaces,
respectively. Thus, the path-connected space
is an
-space
if and only if the canonical mapping
,
adjoint to the identity mapping of
,
admits a left homotopy inverse and the path-connected space
is a
co-
-space
if and only if the canonical mapping
,
adjoint to the identity mapping of
,
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
.
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
-connected
-cell
CW-complexes
,
and a sphere
such that
but
.
This theorem dualizes straightforwardly to an example of the failure
of
product cancellation;
there exist
-connected,
2-stage Postnikov systems
and an Eilenberg–MacLane space
such that
but
.
The much more delicate question of failure of product cancellation
for
-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
-space
(indeed, a loop space) not homotopy equivalent to any of the
"classical"
-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
-spaces;
these would be finite Postnikov spaces which are
co-
-spaces.
Since a non-contractible,
-connected,
finite
Postnikov space cannot be a
co-
-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
,
the co-product of finitely many circles, is the only
path-connected finite Postnikov
space admitting a
co-
-space
structure. Thus the dual of a spicy piece of homotopy theory can be
rather bland.
The space
calls to mind another example, discussed in
[a2],
and given here. If
is a path-connected
-space
of
finite homotopical type
(all homotopy groups are finitely generated), then there exists an
-space
with
and a homotopy equivalence from
to
.
Dually, if
is a path-connected
co-
-space
of
finite homological type
(all homology groups are finitely
generated), there exists a
-connected
co-
-space
and a mapping from
to
inducing homology isomorphisms. In
1971,
Ganea posed the question of whether
is homotopy equivalent to such a co-product. Recently
(1997),
N. Iwase
has announced a negative answer to this
question.