Continuous operations on a topological space
by
Walter Taylor (Boulder, CO)
These are notes for my talk on August 16, 2004. It contains a bit more than I
shall be able to say then. There is no legitimate way to archive it or to reference
it. For citation, please use [8] or [9].
1
Introduction: the compatibility relation.
For a topological space A, and a set Σ of equations of any similarity type,
the relation
A |= Σ
(A is compatible with Σ), is taken to mean that there are continuous operations on A such that, taken with these operations, A is a model of Σ. Thus,
for example, if Γ axiomatizes group theory, then A |= Γ means that A is the
underlying space of some topological group.
The most haunting single thing I learned of during my years in graduate
school (mid-sixties)—not that I ever studied it in any official context—was
the richness, or let us say mysterious power, of the compatibility relation
|=. In this talk I wish to explore this richness, and to illustrate it with two
recent results (Theorems 1 and 2 on page 2), in which we see |= reflecting
the richness of two other areas of mathematics: algebraic topology and the
theory of non-recursive sets.
The power of |= is immediately obvious: if A is e.g. a compact real interval, then A is not the underlying space of a topological group, and nobody
is going to make A into a topological group, ever, period! (Moreover, this
example appears to partake of a general strictness: A |= Σ appears to be true
with probability zero.1 ) As for the mystery, suffice it to say that, from even
very few basic examples in the area, one starts to notice two things. One
is a lack of discernible pattern, the lack of simple rule describing |=. The
1
It remains open to state this conjecture in a mathematically meaningful way, and to
prove it.
1
other is that all known results—whether of the form A |= Σ or of the form
¬(A |= Σ)—seem to be discovered and proved ad hoc.
Like any relation, we may regard |= as a potentially infinite chart; we
illustrate with a dozen values of the relation. The interested reader may try
to work out these values for himself (using Theorem 1 for S 2 ); this exercise
should adequately illustrate the ad-hoc nature of proof in this area.
R
S2
A···
no
yes yes
no
···
lattice theory
yes
no
yes
no
···
bounded lattice theory
yes
no
no
no
···
Σ
..
.
..
.
..
.
..
.
..
.
|=
[a, b]
group theory
S1
..
.
With a few notable exceptions, it is a formidable task to reach any clear
conclusion about any single row of this chart. (Topological groups are themselves a lifetime study, and for e.g. topological lattices, the study has not yet
begun.) It has recently proved more profitable to study a single column of
the chart; that is, to examine A |= Σ for a fixed space A. In this talk we are
concerned with A = S n (the n-sphere) and A = R.
Here are the two recent theorems that I mentioned above. (Undemanding
means trivial, in a sense to be made precise in §2.4.)
Theorem 1 (W. Taylor [8].) (n 6= 1, 3, 7.) If Σ is compatible with S n , then
Σ is undemanding.
Theorem 2 (W. Taylor [9].) There is no algorithm that decides, for each
finite set Σ of equations, whether Σ is compatible with R.
These two theorems do not completely dispel the mysterious quality of the
relation |=, but they do help correlate it with some other rich and mysterious
aspects of mathematics, which are perhaps more familiar. Theorem 1 locates
part of our mystery squarely among the intricacies of algebraic topology (the
special role of S 1 , S 3 and S 7 ); Theorem 2 locates part of it squarely in the
2
realm of non-recursive sets. (And thus it is no wonder that proofs in this
area are discovered ad hoc.)
It turns out that Theorems 1–2 are relatively easy to prove, if one uses just
the right tools from algebraic topology and from decidability theory. In fact,
all the tools for Theorem 1 were in place by 1963, when I first contemplated
these issues; so in principle I could have proved it then. (What I gained in the
meantime was confidence in working with identical satisfaction.) Our proof
of Theorem 2 has been available at least since 1970, when Ju. V. Matiyasevich
proved the undecidability of diophantine equations [5]. We shall skim over
the proofs in an attempt to highlight their essential simplicity.
2
Laws on spheres — proof of Theorem 1.
We shall approach Theorem 1 in such a way as to highlight the fact that
it is a straightforward and easy generalization2 of its better-known classical
special case: for these values of n, S n is not an H-space (Theorem 5 below).
H-spaces are defined3 by the Σ that contains these two laws:4
F (x, e) ≈ x;
2.1
F (e, x) ≈ x.
(1)
The vector degree of an operation on spheres.
Consider a continuous k-ary operation on S n ,
F : (S n )k −→ S n .
(2)
The degree of F is a vector of integers (d1 , · · · , dk ). We will not define it
here, but will instead list its properties that we need:
• The degree is an invariant of the homotopy class of F .
• The degree of a constant map is (0, · · · , 0).
2
This is not exactly the way I discovered it.
An equation σ ≈ τ , with the wavy equal-sign, is a syntactic object; it makes no
assertion, but merely puts forward a pair terms for consideration. The assertion, if any,
lies in words like “axiomatizes,” or “defined by,” or in the satisfaction relation |=.
4
Theorem 5 is usually extended to satisfaction within homotopy; we ignore this embellishment, except to say that of course Theorem 1 remains valid in the homotopy context
as well.
3
3
• The ith projection map has degree (0, · · · , 0, 1, 0, . . . , 0), with 1 in the
ith co-ordinate.
• The degree of a composite map may be calculated from a simple bilinear formula, namely the formula (3) that appears in the following
lemma.
Lemma 3 Suppose that F : (S n )k −→ S n has degree (e1 , · · · , ek ), and that
j
Gj : (S n )N −→ S n has degree (g1j , · · · , gN
) (1 ≤ j ≤ k). Then the composite
n N
n
map H : (S ) −→ S , defined by
H(x) = F (G1 (x), · · · , Gk (x))
(for x ∈ (S n )N ), has degree (d1 , · · · , dN ), where
di =
k
X
ej gij .
(3)
j=1
Incidentally, the categorical import of Lemma 3 is as follows. Let Cn denote the full subcategory of topological spaces, whose objects are the powers
(S n )k for k ∈ ω. For each map G : (S n )N −→ (S n )k , let D(G) denote the
N × k matrix whose ith column is the degree of πi ◦ G (1 ≤ i ≤ k). The
lemma may then be interpreted as asserting that D is a product-preserving
functor (i.e. a morphism of abstract clones) from Cn to the category of all
rectangular matrices of integers (which is the abstract clone associated to
Abelian group theory).
We first note that, for n = 1, 3 or 7, the multiplication of unimodular
complex numbers, quaternions or Cayley numbers, respectively,
(x1 , · · · , xk ) 7−→ (· · · ((x1 x2 )x3 ) · · · )xk
(4)
has degree (1, · · · , 1). A major result of algebraic topology states that, for
other values of n, there can be at most one odd component.
Theorem 4 If n 6= 1, 3, 7, and
F : (S n )k −→ S n ,
(5)
is a continuous k-ary operation of degree (d1 , · · · , dk ), then at most one di is
odd.
We return in §2.6 to a discussion of Theorem 4, its proof, and its place in
the literature; for the moment, let us complete our simple proof that there
is no H-space based on S n for n 6= 1, 3, 7.
4
2.2
Non-existence of H-spaces on spheres.
Theorem 5 (Hopf, Adams, et al.) Suppose that we have a constant e ∈ S n
and a continuous binary operation F : S n × S n −→ S n such that (S n , F , e) is
an H-space. Then n = 1 or 3 or 7.
Proof.
Let the degree of F be (d1 , d2 ). From the law F (x, e) ≈ x, we
immediately see that d1 = 1; similarly d2 = 1, and the result is immediate
from Theorem 4.
Notice that this proof did not require us to make any use of our fourth
property (Lemma 3). (It will be used in §2.3 and in §2.4.) Notice also
that, by the first property, the same proof actually rules out H-spaces up to
homotopy. This extension also holds for results in §2.3 and in §2.4 (but will
not be further mentioned).
2.3
A more complicated proof for Theorem 4.
For the sake of generalization, we complicate our proof of Theorem 5 as
follows. Suppose as before that a constant e ∈ S n and a continuous F :
S n × S n −→ S n yield an H-space (S n , F , e). We are then looking for a
contradiction. By Theorem 4, at most one of d1 and d2 is odd. Suppose we
define a new operation F 0 : S n × S n −→ S n to be first-coordinate projection if
d1 is odd, and second-coordinate projection if d2 is odd. (Let it be constantly
e if both are even.) From the H-space law F (x, e) ≈ e, we see that F (x, e) is
the identity function of x, and hence by Lemma 3 that d1 is odd, and hence
(by the definition of F 0 ) that
(S n , F 0 , e) |= F (x, e) ≈ x.
A similar argument establishes that d2 is odd5 and hence that
(S n , F 0 , e) |= F (e, x) ≈ x,
and hence that (S n , F 0 , e) is an H-space. One immediately checks, however,
that neither a projection operation nor a constant, such as F 0 , can be the
5
The astute reader will notice that we already have one contradiction, namely that d1
and d2 are both odd. However, that is not the argument that will generalize, and it is
nowhere written that one must always take the first contradiction that comes along. In
any case, what we are doing here is merely motivational for §2.4.
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binary operation of an H-space. This contradiction completes the alternate
proof of Theorem 5.
The key element here was that from(S n , F , e) an H-space, we got (S n , F 0 , e)
to be an H-space. This is the part that will be repeated for arbitrary Σ.
2.4
Generalization to arbitrary laws.
A careful examination of the proof in §2.3 provides us both the right definition, and the beginnings of the right proof, for the best extension to an
arbitrary equational axiom system Σ. We call Σ easily satisfied or undemanding if there exists a model (A, Ft0 )t∈T of Σ in which |A| > 1 and for which
each Ft0 is either a coordinate projection or a constant operation.
A moment’s reflection should convince the reader that every undemanding
theory can be modeled with continuous operations on S n (or on any space,
for that matter). The converse statement is the correct generalization of
Theorem 5; moreover its proof is a straightforward generalization of the proof
we presented in §2.3.
Remark. There is an easy algorithm to check if a finite Σ is undemanding. It may therefore be regarded as obvious, or essentially known, that the
equations defining H-spaces are not undemanding. It is therefore obvious
that the classical Theorem 5 is a special case of Theorem 1.
2.5
Proof of Theorem 1.
We are given continuous operations Ft : (S n )n(t) −→ S n such that
(S n , Ft )t∈T |= Σ
(6)
and we need to construct operations Ft0 modeling Σ, each a constant or a
projection.
In fact B can be taken as any set with more than one element. We then
let c be any element of B, and define the operations Ft0 as follows:
(A) If the degree of Ft is (d1 , · · · , dn(t) ), with each di even, then
Ft0 (x1 , · · · , xn(t) ) = c.
(B) If di is odd, then
Ft0 (x1 , · · · , xn(t) ) = xi .
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(Of course, in clause (B), only one di can be odd, by Theorem 4 above.) It
remains only to show that
(B, Ft0 )t∈T |= Σ.
(7)
Let τ be a term in the operation symbols Ft , and let N be an integer
greater than n for each xn appearing in τ . (Such N will be called appropriate.)
We let τ N denote the realization of τ as a (continuous) N -ary operation on
S n , formed by letting each operation symbol Ft be realized by F t . (It has
a well-known recursive definition, which we skip.) Similarly, τ 0N denotes an
operation that is similarly constructed on B, using the operations Ft0 .
It is well known and easy to show inductively that (S n , Ft )t∈T |= σ ≈ τ
if and only if σ N = τ N (for appropriate N ). Similarly (B, Ft0 )t∈T |= σ ≈ τ
if and only if σ 0N = τ 0N . The next lemma generalizes (A) and (B) above
to terms. We omit the straightforward proof, which uses induction on the
length of τ , and the four properties of the vector degree that are listed at the
start of §2.1.
Lemma 6 For any term τ and appropriate N , if the degree of τ N is (d1 , · · · ,
dN ) with each di even, then τ 0N (x) = c for any x ∈ B N . If some di is odd,
then τ 0N (x) = xi for any x ∈ B N .
We now complete the proof of Theorem 1, by proving (7). Suppose σ ≈
τ ∈ Σ. By (6) we have (S n , Ft )t∈T |= σ ≈ τ , and so σ N = τ N for appropriate
N . It follows immediately from Lemma 6 that σ 0N = τ 0N , and hence that
(B, Ft0 )t∈T |= σ ≈ τ . Thus (7) is proved and the proof is complete.
2.6
The real story.
So why did it take me thirty years to find these results? If there is any answer
besides my own natural slowness, it may be the fact—which I have hitherto
concealed—that Theorem 4 is not actually in the literature.
The usual treatment concerns an integral invariant Γ(G) of a continuous
map
G : S 2n+1 −→ S n+1 .
(8)
Γ(G) is known as the Hopf invariant of G, introduced by H. Hopf in 1935
[3]; for alternate definitions see Dieudonné [2, pp. 314–317] or Hu [4, pp.
334–335]. Now although Theorem 4 does not appear in the literature; it is
not terribly hard to see (we skip the details) that it follows from the following
two theorems:
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Theorem 7 If G as in (8) has odd Hopf invariant Γ(G), then n = 1, 3 or 7.
Theorem 8 For any binary operation F : S n ×S n −→ S n , there is a mapping
G as in (8) such that
Γ(G) = ± d1 d2 ,
(9)
where (d1 , d2 ) is the degree of F .
Theorem 7 is one of the crowning achievements of algebraic topology. It
was proved in stages by G. W. Whitehead (1950; n ≡ 1 (mod 4)), by J.
Adem (1956; n 6= 2k − 1), by H. Toda (n = 15), and finally by J. F. Adams
(1958; all n 6= 1, 3, 7). See e.g. Dieudonné [2, pp.549–551] for a discussion of
the various proofs. We make no claims of improving either the statement or
the proof of this result. We merely make use of it, in an essential way.
Theorem 8 is less often seen in the literature, especially without the assumption that F is an H-space operation. We have taken this version from
page 13 of Steenrod [7].
To recapitulate: had Theorem 4 been prominent in the literature, our
Theorem 1 might have been easily available back in the late sixties when
I first started looking at these questions. Moreover, had Theorem 4 been
prominent in the literature, the correct notion of triviality, (namely, that of
an undemanding Σ), would have been more evident from the start, as we saw
with our second proof of Theorem 5 in §2.3.
2.7
Viewpoint.
In the right context, therefore, Theorem 1 may be seen as a straightforward
generalization of Theorem 5. Nevertheless, to a general algebraist, Theorem
1 is clearly the right theorem—the one that ought to be better known—and
Theorem 5 (about H-spaces) should properly take on the role of a special
case.
I think I may say this without any particular boast for my own modest
contribution here. It seems simply to me that, had a more general algebraic
viewpoint held sway earlier, then Theorem 1 is what would have been proved
by our ancestors, long before I came along. The original topologists reached
for the best generalization they could find, for “not a topological group.”
What they came up with was, “not an H-space.” It is now clear that, at
least in the realm of equational theories, the best possible generalization is
this: “not a topological model of any demanding theory.”
8
3
Laws on R — proof of Theorem 2.
3.1
Satisfaction of Diophantine equations.
We understand a Diophantine equation to be an equation p ≈ q for algebras
hA; +, ·, −, 0, 1, ci ii∈ω of type h2, 2, 2, 1, 1, 0, 0, . . .i. An integral solution for
p ≈ q, or a solution in Z is an ω-tuple hc0 , c1 , . . .i of integers such that
hZ; +, ·, −, 0, 1, c0 , c1 , . . .i |= p ≈ q.
(10)
(Clearly the truth of (10) depends only on finitely many ci , namely, those ci
for which ci appears in p or in q.) A Diophantine equation p ≈ q is said to
be Z-satisfiable, or satisfiable (or solvable) in integers, or sometimes—if the
Diophantine context is clear—simply solvable or satisfiable, iff there is some
ω-tuple hc0 , c1 , . . .i such that (10) holds.
(Our present definition of the satisfaction or solvability of Diophantine
equations differs slightly from the classical presentation. Normally, the unknowns would appear as variables xi rather than as nullary function symbols
ci . However, as the reader will easily see, the present usage captures the same
notion of satisfiability. We have adopted it because it meshes more directly
with other notions of satisfiability in the paper. Specifically, its use simplifies
the proof of Theorem 2.)
3.2
A Σ associated to each Diophantine equation.
For each Diophantine equation p ≈ q, we form the set of equations
Σp,q = Σ0 ∪ Σ1 ∪ {s(2ci ) ≈ 0 : ci appears in p or in q} ∪ {p ≈ q},
where
A finite set of equations in +, ·,
−, 0, 1, defining commutative ring
theory with unit.
Σ1 =
c(x + y) ≈ (c(x) · c(y)) − (s(x) · s(y)),
s(x + y) ≈ (c(x) · s(y)) + (s(x) · c(y)),
c(s(x)) ≈ λ(x) · λ(x),
c(0) ≈ 1, c(1) ≈ 0, s(1) ≈ 1 .
Σ0 =
9
(11)
(12)
(13)
(14)
At this stage there is nothing to prove. §3.2 merely states a definition, and
a syntactic one at that.
3.3
Models of Σp,q .
Provided that the Diophantine equation p ≈ q is Z-solvable, one R-model of
Σp,q is easy to describe. We shall describe this model and then prove that
every topological R-model is bi-continuously isomorphic to one of this type.
Suppose that we have integers ci that satisfy p ≈ q, i.e., such that (10)
holds. Since Z ⊆ R, we clearly also have
hR; +, ·, −, 0, 1, c0 , c1 , . . .i |= p ≈ q.
(15)
It is of course well known that
hR; +, ·, −, 0, 1, c0 , c1 , . . .i |= Σ0 .
Finally, if we define c and s to be the continuous functions
s(x) = sin(
πx
);
2
c(x) = cos(
πx
),
2
and make the obvious definition for λ, then it is not hard to see that
hR; +, ·, −, 0, 1, c, s, λ, c0 , c1 , . . .i
satisfies Σ1 and moreover satisfies s(2ci ) ≈ 0 for each i. In other words, we
now have
hR; +, ·, −, 0, 1, c, s, c0 , c1 , . . .i |= Σp,q .
Lemma 9 Every topological algebra based on R that models Σp,q is bi-continuously isomorphic to one of the type just described.6
Proof.
(Sketch.) Suppose that
A = hR; , , , 0, 1, c, s, λ, c0 , c1 , . . .i |= Σp,q
(16)
with , and continuous functions of two real variables, c, s and λ continuous functions of one variable, and with the remaining operations nullary
(constants).
6
Our description included an ambiguity of sign in the definition of λ; the intent of the
lemma is that for some choice of this sign, we will have isomorphism.
10
One begins by showing—using the topology of R and the axioms Σ0 —
that there is a bi-continuous isomorphism φ between hR; , , , 0, 1i and
the standard structure hR; +, ·, −, 0, 1i. It is a classical result going back to
Cauchy [1], that for continuous c and s, Equations (11–12) imply either that
φ c φ−1 and φ s φ−1 are the obvious trigonometric functions
φ c φ−1 (x) = cos(αx)
φ s φ−1 (x) = sin(αx),
(17)
(18)
for some α ∈ R, or that they are both constantly zero. The latter possibility
is ruled out by (14), and so we may consider that Equations (17–18) are true.
It follows fairly directly from Equations (13–14) that α = π/2. We leave it
to the reader to sort out the situation with λ.
If c0 , c1 , . . . are the constants from (16), it is easy to see that φ(c0 ), φ(c1 ), . . .
are real numbers satisfying the original equation p ≈ q. In order to have a
model of the type described above, we must have that each φ(ci ) is an integer.
This is immediate from Equation (18) and the fact that A |= s(2ci ) ≈ 0.
Corollary 10 A Diophantine equation p ≈ q is Z-satisfiable iff R |= Σp,q .
3.4
Proof of Theorem 2.
The proof is by contradiction. Suppose we had an algorithm A to decide
whether Σ is compatible with R. It is then immediate from Corollary 10
that the following algorithm A0 will decide whether an arbitrary Diophantine
equation p ≈ q is Z-satisfiable: given an equation p ≈ q, A0 simply constructs
Σp,q and passes it to A for an answer. On the other hand, it is known that
no such algorithm A0 exists (Ju. V. Matiyasevich [5, 6], building on the work
of J. Robinson, M. Davis, et al.). This contradiction completes the proof of
Theorem 2.
References
[1] A.-L. Cauchy, Cours d’Analyse, Sér. 2, 3 (1887), 106–113.
[2] J. Dieudonné, A History of Algebraic and Differential Topology 1900–
1960, Birkhäuser, Boston, 1989. QA 612 D54 1989
11
[3] H. Hopf, Über die Abbildungen von Sphären auf Sphären niedrigerer
Dimension, Fundamenta Mathematicae 25 (1935), 427–440.
[4] S.-T. Hu, Homotopy Theory, Academic Press, New York, 1959. QA3 .P8
volume 8
[5] Ju. V. Matiyasevich, Diofantovost’ perechislimykh mnozhestv. Doklady
Akademiı̆ Nauk SSSR 191 (1970), 297-282 (Russian). English translation: Enumerable sets are Diophantine, Soviet Mathematics Doklady 11
(1970), 354-358.
[6]
Hilbert’s Tenth Problem, The MIT Press, Cambridge, London,
1993.
[7] N. E. Steenrod, Cohomology Operations, Annals of Mathematics Studies,
volume 50, Princeton University Press, 1962.
[8] W. Taylor, Spaces and equations, Fundamenta Mathematicae 164
(2000), 193–240.
[9]
Equations on real intervals, preprint, 2003.
Walter Taylor
Mathematics Department
University of Colorado
Boulder, Colorado 80309–0395
USA
Email: [email protected]
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