Incompatibility of algebra and topology Walter Taylor Mathematics Department University of Colorado Boulder, Colorado 80309-0395 USA January 11, 1999
Contents 0 Introduction.
3
1 Terms, equations and satisfaction. 1.1 Terms and interpretations. . . . . . . . . . . . . . . . . . . . . . . 1.2 Satisfaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 5 7
2 Identical satisfaction in topological spaces.
8
3 Statement of the main results. 3.1 Undemanding sets of equations. 3.2 The main theorems. . . . . . . 3.3 q th powers. . . . . . . . . . . . 3.4 A lemma for all the proofs. . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
9 9 11 13 14
4 The path groupoid of a topological space.
15
5 The path groupoid of a topological algebra
17
6 Free Groups.
19
7 The proof of Theorem 1.
23
8 Commutative graded rings; tensor products. 26 8.1 The essential facts. . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.2 Some technical lemmas on CGR’s. . . . . . . . . . . . . . . . . . 28 9 The cohomology ring of a topological space.
1
33
10 The 10.1 10.2 10.3 10.4 10.5 10.6
proofs of Theorems 2–6. The co-operations Ft? and their properties. Even-dimensional spheres. (Theorem 2.) . . The orientable surface of genus 2. (Theorem The Klein bottle. (Theorem 4.) . . . . . . . Real projective space. (Theorem 5.) . . . . The figure-eight. (Theorem 6.) . . . . . . .
. . . . 3.) . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
35 35 38 40 41 42 44
11 Degrees and the Hopf invariant.
46
12 The proof of Theorem 7
50
13 A sketch of the proof of Theorem 8.
52
14 Final remarks and problems. 14.1 Satisfaction up to homotopy. . . . . . . . . . . 14.2 Generalizations and extensions. . . . . . . . . . 14.3 Our results, interpreted in clone theory. . . . . 14.4 The lattice of interpretability. . . . . . . . . . . 14.5 A construction for topological algebras. . . . . 14.6 Compatibility and H-spaces. . . . . . . . . . . . 14.7 Compatibility and commutativity of homotopy.
2
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
53 53 54 54 55 59 59 60
This preprint has been made available by the author, both for the information and convenience of fellow scholars, and for the author to establish his priority in these discoveries. Subject to final corrections and improvements, the material will in due course be published by the author through normal channels. The author retains any and all copyrights to this material. Limited permission is granted to individual scholars to obtain individual paper copies in order to advance their study of the material. Limited permission is granted, until revoked, to web sites at Vanderbilt, Nipissing and Purdue Universities, to make available a postscript version of the manuscript. Permission is not granted for posting on any other sites, including mirrors or any other sort of derivative site. In no case is permission granted for possession of source-code for this material. The author would of course be glad to learn of anyone’s discoveries concerning this material, including any typos or errors, and any pointers to related material. Send mail to
[email protected].
Incompatibility of algebra and topology by Walter Taylor
0
Introduction.
There is a long history of the idea of compatibility between a topological space A and a set Σ of equations. By compatibility, we mean the possiblity of modeling Σ by continuous operations on A. For example, if Σ axiomatizes group theory, then we are referring to the possibiity of making A into a topological group. Historically, algebraic topology has derived much of its motivation from attempts, ulitmately very successful, to prove certain nowadays well-known incompatibility theorems. Notably, it was proved (see §11 below for a brief history, and some references) that, except for n = 1, 3, 7, the n-sphere S n is incompatible equations Σ axiomatizing group theory. (In fact, S n is incompatible with a much weaker Σ, namely the equations defining H-spaces — see Equations (16) in §3.2 below.) 3
In this article we extend these results to arbitrary Σ, Thus, for the classically important spaces A = S n (and for various other A), we prove that A is incompatible with all Σ (except certain obviously trivial Σ that are described in §3.1 below). The advances described here lie in the domain of general algebra and equational logic; algebraic topology, while it is applied in an interesting manner, is not really advanced in its own right. The precise statements of our theorems are in §3.2. This is in fact one in a series of articles by the author [40] [41] [42] concerned with the (universal) satisfaction relation (A, F t )t∈T |= σ ≈ τ
(1)
between an equation σ ≈ τ and a topological algebra (A, F t )t∈T . (The universe A of the algebra is a topological space A = (A, T ), and each operation F t is assumed continuous as a map defined on a finite power of this space. For example, we might have n(t) = 2; in that case F t is a continuous binary operation on A. We rarely need to mention directly the topology T , but we will need occasionally to invoke the continuity of the F t . Here σ and τ are terms, i.e. formal expressions involving variables, such as the two sides of the well-known associative law. Informally then, Relation (1) says that the expressions σ and τ come out the same when interpreted by the operations F t . In the example just mentioned, (1) says that Ft is associative. The reader should refer to 1.2 for a complete definition of (1).) We also consider the extension of (1) to a set Σ of equations. In other words, we consider (A, F t )t∈T |= Σ, (2) which means that (1) holds for every equation σ ≈ τ in Σ. Our main results — precisely stated in §3.2 below — concern some very simple spaces: the figure-eight space (i.e. two circles joined at one point), spheres of dimension 6= 1, 3, 7, the orientable surface of genus 2, the Klein bottle, and real projective spaces of dimension 6= 1, 3, 7, 15, · · · . If A is one of these spaces (or in some cases, like one of them in homotopy or cohomology), then there are no continuous operations F t on A satisfying (2) unless Σ is so trivial that it can already be modeled with constants and projections maps. The proofs will be presented in §7, §10 and §12. There was a long-standing problem, here solved negatively, of finding a Σ that is non-trivial in this sense, but also modelable on one of these spaces. The problem probably first appeared in print in 1986, when it was raised (for a number of spaces) on page 38 of [42].1 1 Problem 9.4 on page 83 of [42] in fact points out that for almost any space (A, T ) that one can name, it is open whether (2) holds for any non-trivial Σ. This article represents the first time that we can answer the question negatively for any relatively simple space (A, T ). I am happy to say also that Problem 9.1 [loc. cit.] has been solved affirmatively by Vera Trnkov´ a. There do exist spaces A and B that are the same at the first clone level, but whose
4
The results here work a little differently than those found in 1977 [40], in 1981 [41] or in 1986 [42]. In those earlier articles, some general properties of the space (A, T ) (such as non-commutativity of its fundamental group) were used to rule out the satisfaction relation (2) for certain Σ’s (such as Σ defining group theory). In this article, we narrow our focus down almost to a single space (A, T ) (e.g. by specifying the isomorphism type of its fundamental group), and then prove that in this context (2) fails for all Σ (except for certain almost trivial Σ). This different approach obviously requires more attention to details about an individual space or group. We attend to these details concerning free groups (some easy extensions of known results) in §6, and concerning tensor products in §8. It is our experience in speaking about this material that few mathematicians are conversant in both the requisite equational logic (§1) and the requisite algebraic topology (§4, §8, §9, §11). Those who are conversant in either of these subjects will find the corresponding sections somewhat elementary, and can obviously move on, after possibly learning our notation. In spite of the requisite attention to detail, the main spirit of the paper is still categorical. We rely heavily on the functorial properties of the homotopy group and the cohomology ring. See also the final remarks in §14.3, where we give an alternate explication of our results in terms of abstract clone theory (algebraic theories), which is a branch of category theory. The author thanks the University of Colorado for their partial support of his sabbatical leave of 1997–98, during which time some of this material was developed. He thanks the members of the University of Hawaii seminar in lattice theory and general algebra (especially Ralph Freese, Bill Lampe and J B Nation), who generously gave of their time in March of 1998 to hear a full proof of of Theorem 2. He also thanks George McNulty, Jan Mycielski and Stanislaw ´ Swierczkowski, who listened to full proofs of Theorems 1 and 2 in October of 1998.
1 1.1
Terms, equations and satisfaction. Terms and interpretations.
The material of this section is elementary but subtle. In particular, we need to distinguish carefully between a term (symbolic composite operation) τ and various composite operations such as τ , τ ? and τ 0 that are patterned after τ . The operations τ are essential to a precise understanding and a precise mathematical definition of the identical satisfaction relation (1); and the recursive construction of τ , τ ? and τ 0 is essential to the inductive arguments that are needed in our clones satisfy different first-order sentences at higher levels. See recent articles by V. Trnkov´ a [44] [45] [46] [47], and by J. Sichler and V. Trnkov´ a [34].
5
proofs. The reader who is familiar with this material can read quickly, while pausing to take in our notation; although a little fussier than usual, it is essential to the remainder of the article. To express these notions carefully, we begin with an indexed collection of function symbols Ft (t ∈ T ), to stand for operations on a set A. Attached to each t ∈ T is a non-negative integer n(t) called the arity of Ft . An interpretation of Ft on a non-empty set A is an n(t)-ary operation on A, i.e., a function Ft : An(t) −→ A. (Thus when we say “The operations F t interpret the function symbols Ft ,” the only real assertion is that each F t has the correct domain An(t) .) In many cases of interest, there are only one or two operations Ft , having traditional designations like +, ·, ∧, ∨, etc. We will use these familiar designations when they are available. Sometimes one omits the bar from Ft , allowing the context to differentiate the symbolic operation from the concrete operation; this practice is especially widespread in the case of +, ∧, etc. A term is a symbolic expression that is recursively defined to be either a variable xi (for some i = 0, 1, 2, . . . ), or Ft (τ1 , . . . , τn(t) ) for some t ∈ T and some simpler terms τj . An equation is an ordered pair of terms (σ, τ ). This pair is usually written σ ≈ τ , with the bent equal-sign emphasizing the role of equality in the interpretation of σ ≈ τ , which we describe presently. Nevertheless, it should be remembered that “σ ≈ τ ” merely symbolizes an equation as a linguistic entity; by itself it makes no assertion. On the other hand, “σ = τ ” does make an assertion: it asserts that σ and τ are precisely the same term. Our proofs about terms are usually by induction. One way to say this is that we induct over the well-founded order defined on the set of all terms by always taking τj to lie below Ft (τ1 , · · · , τn(t) ). A more elementary plan — which we adopt — is to assume we have |τ | ∈ ω for every term τ , with |τj | always less than |Ft (τ1 , · · · , τn(t) )|, and then to carry out an elementary inductive proof relative to the quantity |τ |. There are many possible ways to define |τ |, such as the number of function symbols in τ . If the operations Ft interpret the symbols Ft on a set A, then every term τ has an associated interpretation τ : Aω −→ A which is defined2 recursively on A via xi (a) τ (a)
= =
ai Ft (τ1 (a), . . . , τn(t) (a))
(3) (4)
where τ = Ft (τ1 , . . . , τn(t) )
(5)
2 Recall that ω = {0, 1, 2, 3, . . .}. We adopt the convention that if a ∈ Aω , then a denotes i the ith component of a. In other words, a = ha0 , a1 , a2 , . . . i.
6
Notice that the bar notation is not essential to the construction described in Equations (3) and (4). In §5 we will interpret the function symbols Ft with operations Ft? , and in §7 and §10 with function symbols Ft0 . In those contexts, each term τ will have corresponding interpretations τ ? and τ 0 . In §10 we will need the N -restricted interpretation τ N : AN −→ A. This will be defined only when the variables appearing in τ are among {xi : i < N }. In fact τ N is also defined by Equations (3–5) that define τ , but with the domain changed to AN . In §10 it will be helpful to have Equation (4) recast as τ N = Ft ◦ τbN
(6)
where τbN : AN −→ An(t) is specified by the equations n(t)
πi
◦b τ N = τi N
(7)
n(t)
for 1 ≤ i ≤ n(t). (Where πi denotes the ith coordinate projection from An(t) onto A.) It will also be useful to be able to compare τ N and τ M for M > N . For this purpose, we use the N -fold projection operations M ΠM −→ AN N :A ω ΠN : A −→ AN
which are defined by πiN ◦ ΠN πiN ◦ ΠM N
= =
πi πiM
(8)
th for i < N . (In other words (8) says that the ith component of ΠM N (a) is the i component of a.) It seems obvious that
τ N ◦ ΠN τ N ◦ ΠM N
= =
τ τM
(9)
for M > N , and moreover these equations have an easy inductive proof involving (6) (which we omit). One easily proves by induction that τ (a) (or τ N (a)) depends only on the variables appearing in τ , i.e., that τ (a) = τ (b) if ai = bi for each i ∈ ω with xi appearing in τ . If these variables are xi0 , xi1 , . . . , one sometimes writes τ (ai0 , ai1 , . . .) in place of τ (a).
1.2
Satisfaction.
An interpretation Ft (t ∈ T ) on A is said to model or identically satisfy an equation σ ≈ τ iff σ = τ (as functions defined on Aω ). (The word “identically” 7
can be be omitted in a context such as this one, where identical satisfaction is the main topic. The reader is, however, advised that, in general, satisfaction is a more elaborate topic.) Satisfaction has the notation (A, Ft )t∈T |= σ ≈ τ,
(10)
which relates the set A, the operations Ft , and the formal terms σ and τ . Sometimes we say instead that σ ≈ τ is an identity of (A, Ft )t∈T . For a set Σ of equations, we write (11) (A, Ft )t∈T |= Σ, and say that the interpretation Ft (t ∈ T ) models or identically satisfies Σ, if (A, Ft )t∈T |= σ ≈ τ for every equation σ ≈ τ ∈ Σ. A tuple of the form (A, Ft )t∈T , i.e., a set with operations, is an algebra, and so quite often one reads (11) as saying that the algebra (A, Ft )t∈T models, or satisfies identically the equations Σ. That terminology is less useful in this paper, since our main point is to prove that no interpretation models Σ. In this article we work with formal equations σ ≈ τ in only one way: we invoke the definition of satisfaction to obtain σ = τ for some interpretation Ft (t ∈ T ). Thus no formal work (deductions, etc.) is required for ≈. It should be obvious that if |A| = |B|, and if (A, Ft ) |= Σ, then there are operations Gt on B that also model Σ. Moreover, if Σ has any model of more than one element, and if |A| ≥ |T | + ℵ0, then there are operations3 Ft on A that also model Σ. Thus, for sets A, the existence of models is mostly independent4 of the choice of A. As we now turn to topological spaces, we will see that situation change radically.
2
Identical satisfaction in topological spaces.
If A is (the underlying set of) a topological space,5 we may ask whether the operations Ft : An(t) −→ A are continuous (with respect to the usual product topology on A). If they are, and if (A, Ft )t∈T |= Σ, then we say that the operations Ft model Σ continuously on A. (We may also say that (A, Ft )t∈T is a topological algebra satisfying Σ.) If there are any operations Ft continuously modeling Σ on A, then we say that Σ is continuously modelable (satisfiable) on the space A, or that A supports 3 One could invoke the L¨ owenheim-Skolem Theorem for this fact. It is simpler to construct (A, Ft ) as a subalgebra of a power of the given model with more than one element. 4 The obvious exception to this statement concerns models (A, F ) t t∈T with |A| < |T | + ℵ0 . The set of all such |A| is known as the spectrum of Σ, and is a non-trivial topic of investigation. See the introduction to [39] for references to equational spectrum problems. 5 In this article, we denote a space and its underlying set by the same letter.
8
Σ, or simply that A and Σ are compatible. Thus, for example, if Σ is a set of equations axiomatizing lattice theory, then Σ is compatible with an interval I = [a, b] (by defining x ∧ y to be the smaller of x and y and x ∨ y to be the larger). On the other hand, I is incompatible with group theory, as shown by the following well-known argument. Suppose that group operations · and −1 were continuously defined on I. Pick c ∈ I with a < c < b, and define f (x) = c · a−1 · x
g(x) = a · c−1 · x.
Then f and g are mutually inverse continuous selfmaps of I, hence auto-homeomorphisms of I. But g takes the cutpoint c of I to the non-cutpoint a. The contradiction establishes the non-existence of continuous group operations · and −1 . It is easily checked that if each equation in Γ is a consequence of Σ — denoted Σ ` Γ — then every space compatible with Σ is compatible with Γ. It follows readily that if Σ1 and Σ2 axiomatize the same equational theory, then Σ1 and Σ2 have exactly the same compatibilities. This remark, should enhance one’s understanding of the material, but is not needed in our proofs. Hence we omit any detailed treatment of the consequence relation.
3
Statement of the main results.
In fact, incompatibility (§2) tends to prevail between a space A and a set Σ of equations. Some compatibilities are known, but individual cases are rare enough to be noteworthy in themselves. Topological groups, for example, are abundant, but occur only on very special spaces.
3.1
Undemanding sets of equations.
Before going on, we should first acknowledge that certain sets Σ of equations are compatible with every topological space A, for trivial and obvious reasons. We call a set Σ of equations easily satisfied or undemanding if, on some set A of more than one element, there are operations Ft : An(t) −→ A, such that (A, Ft )t∈T |= Σ and such that, for each t ∈ T , either (i) There exists i with 1 ≤ i ≤ n(t) such that Ft (a1 , . . . , an(t) ) = ai for all a1 , . . . , an(t) , or 9
(ii) There exists c ∈ A such that Ft (a1 , . . . , an(t) ) = c for all a1 , . . . , an(t) . An operation of type (i) is a projection operation, and an operation of type (ii) is a constant operation. Many important sets of equations are undemanding. For example, semigroups are axiomatized by the single equation (xy)z = x(yz).
(12)
If we interpret xy as x (first-coordinate projection), then both sides of (12) become equal to x. (In this example, we could also take xy to be y or a constant.) This example, by the way, takes nothing away from semigroup theory, since the interesting semigroups are not formed in this manner. For some other examples, the reader might wish to check that the title equations of references [5], [8] and [23] are all undemanding. On the other hand, the three equations (xy)z xx xy
= = =
x(yz) x yx,
(13) (14) (15)
which define semilattice theory, form a demanding set of equations: obviously, (Equation (14) cannot be satisfied by a constant, and Equation (15) cannot be satisfied by a projection operation.) In fact, as the reader may easily check, there is an easy (albeit exponential) algorithm to determine if a finite set Σ of equations is demanding or undemanding. It is easy to construct the finite list of all possible interpretations of the operation symbols as constants or projections. For each such interpretation, one easily checks whether the Σ is satisfied. According to J. Mycielski [private communication] the problem is NP-complete. If Σ is undemanding and A is any space whatever, then Σ is modeled on A by projection and constant operations, which are of course continuous operations. Thus an undemanding Σ is compatible with every space A. The converse also holds: if Σ is compatible with all spaces A, then Σ is undemanding. This follows directly (as we pointed out in [42]) from the existence of a space A, known as the Cook continuum, which satisfies (i) For every finite n, every continuous function An −→ A is either a projection or a constant function. Property (i) obviously yields:
10
(ii) A is compatible only with undemanding Σ’s. Until now, the only known proof of this converse (or even of (ii)), required the extravagant step of first establishing the existence of a space A satisfying the highly unconventional, counterintuitive, and difficult6 condition (i). With the aim of refocusing the material on ordinary spaces — e.g., simplicial complexes, where (i) would be ludicrous — we asked in [42] whether (ii) holds for some simple space A. In particular, we asked whether (ii) holds for A taken to be a figure-eight space or the 2-sphere S 2 . (These two spaces stood out because they had no history of compatibility with any demanding Σ, and because they were known to be incompatible with many Σ’s.) We can thus state our main theorems, which answer problem 9.4 of [42].
3.2
The main theorems.
The full statements of Theorems 1–5 involve homotopy and cohomology, and these concepts of algebraic topology are necessary both for the proofs and for a full understanding of the statements. Nevertheless, the statements can also be understood more immediately: Theorem 1 refers to the figure-eight, Theorem 2 refers to spheres of even dimension, and so on. These somewhat less general versions contain the essence of the results. As for homotopy and cohomology, we will include what we need of their basic properties in §4 and §§8–9. See also §14.1 for further remarks on the role of homotopy in these theorems. Theorem 1 The figure-eight. If A is a path-connected topological space whose fundamental group is isomorphic to the fundamental group of the figureeight space (i.e., free on k generators, with 2 ≤ k < ω), and if A is compatible with Σ, then Σ is undemanding. Theorem 2 Even-dimensional spheres, via co-homology. If A is a pathconnected topological space whose cohomology ring (over some field) is isomorphic to the cohomology ring of an even-dimensional sphere — as presented in Lemma 31 — and if A is compatible with Σ, then Σ is undemanding. Theorem 3 The orientable surface of genus 2. If A is a path-connected space whose cohomology ring (over some field) is isomorphic to the cohomology of the orientable surface of genus 2 — as presented in Lemma 38 — and if A is compatible with Σ, then Σ is undemanding. Theorem 4 The Klein bottle. If A is a path-connected topological space whose co-homology ring (over the prime field of characteristic 2) is isomorphic to the cohomology of the Klein bottle — as presented in Lemma 40 — and if A is compatible with Σ, then Σ is undemanding. 6 The only known method for (i) — expounded by Cook in a series of articles culminating in [10] — has a self-contained exposition occupying 75 pages in an appendix to [33].
11
Theorem 5 Real projective space of dimension 6= 2k − 1. Suppose that n + 1 is not a power of 2. If A is a path-connected topological space whose co-homology ring (over the prime field of characteristic 2) is isomorphic to the cohomology of n-dimensional real projective space — as presented in Lemma 42 — and if A is compatible with Σ, then Σ is undemanding. Theorem 6 The figure-eight, again. If A is a path-connected topological space whose co-homology ring (over a field of characteristic 6= 2) is isomorphic to the cohomology of the figure-eight space — as presented in Lemma 44 — and if A is compatible with Σ, then Σ is undemanding. Theorem 2 tells us that an even-dimensional sphere S 2d is not compatible with any demanding theory Σ. That result of course leaves open the question of Σ-compatibility for odd-dimensional spheres S 2d−1 . In fact it is well known that the group of complex numbers of unit modulus is homeomorphic to S 1 . Similarly, the unimodular quaternions form the space S 3 , and the unimodular Cayley numbers form S 7 . Thus S 1 and S 3 are compatible with Σ = group theory. The unit Cayley numbers are not associative, but their multiplication (denoted here by juxtaposition) satisfies at least the equations 1·x ≈ x·1 ≈ x,
(16)
and clearly Equations (16) are demanding (their topological models are known as H-spaces). Thus any analog of Theorem 2 for odd-dimensional spheres will have to leave out dimensions 1, 3 and 7. It turns out that in fact these are the only dimensions that we have to omit. Theorem 7 Spheres, via the Hopf invariant. If n 6= 1, 3, 7, and if S n is compatible with Σ, then Σ is undemanding. (Notice that Theorems 2 and 7 overlap, in that they both cover the evendimensional spheres. However, Theorem 7 is obviously stronger, and Theorem 2 is apparently more general, since its hypotheses are only about the cohomology ring of A.) Each of Theorems 1–7 decides the question of compatibility for every Σ, and hence is a best-possible incompatibility result for its space A. From other perspectives, e.g. the perspective of fixing a consistent demanding theory Σ and characterizing those spaces compatible with Σ, there is nothing even close to a best-possible result. Theorems 1, 2 and 7 have long been known for idempotent Σ. (A set Σ of equations is called idempotent if the equation Ft (x, . . . , x) = x is a consequence of Σ for each t ∈ T .) The idempotent specialization of Theorem 1 follows immediately from Theorems 3.1 and 5.1 of [40]; the idempotent 12
specialization of Theorem 2 is a special case of Theorem 2.8 of [41]; the idempotent specialization of Theorem 7 is a special case of Corollary 3.2 of [41]. By the same token, the very difficult and unmotivated tensor-algebra calculations on pages 80–85 of [41] are completely outmoded by the easier calculations in this article. It is important to realize that the conclusion to Theorems 1–7 is a conclusion about Σ, and not about any particular operations modeling Σ on A. For example, if Σ consists of the commutative law F (x, y) ≈ F (y, x), then on the sphere S 2 there are many ways to model Σ other than with a constant operation — e.g., F (x, y) = φ(d(x, y)), where d represents Euclidean distance, and φ is any continuous function from R to S 2 . Theorem 2 obviously cannot preclude the existence of this F . Rather, its proof will use the cohomology functor to construct a new operation F 0 from F , in such a way that F 0 is either a projection or a constant, and F 0 also satisfies Σ. (In this case, obviously, F 0 turns out to be a constant.)
3.3
q th powers.
Our final theorem requires a mild extension of the notion of an undemanding Σ. We call a set Σ of equations easily satisfied in q th powers, or q-undemanding, iff, on there is a set A of more than one element, which is a q th power (i.e. A = B q for some set B) and there are operations Ft : An(t) −→ A, such that (A, Ft )t∈T |= Σ and such that, for each t ∈ T , and for i = 1, . . . , q, the composite map π
F
t i A = B q −→ B An(t) = B qn(t) −→
is either a projection or a constant. For an example, consider Σ consisting of the single equation F (F (x, y), F (y, x)) ≈ y.
(17)
Clearly Σ is not undemanding. Nevertheless Σ can be satisfied on A = B 2 by defining7 F ((b1 , b2 ), (b3 , b4 )) = (b2 , b3 ), and so Σ is 2-undemanding. 7 For an elementary, but very rich, exposition of some of the many possible ways to define operations on A = B Q in terms of their components in B—and the associated varieties — see Evans [12].
13
One nice thing about interpretations of this type is that they respect the product topology on B q . That is, if B is a topological space, and A is given the q-fold product topology of B q , then (obviously) Ft is a continuous operation on A. Thus, if Σ is q-undemanding, then Σ is compatible with every space A that is homeomorphic to a direct product B q (for any space B). The converse again follows from the existence of a Cook space: if B is a Cook space and A = B q , then all operations on A are of the desired type; hence any Σ compatible with A is q-undemanding. As in our earlier theorems, we here present simpler q-th power spaces that are compatible only with q-undemanding sets of equations. Theorem 8 If the space A = B q , 1 ≤ q < ω, where B is as in Theorem 1 or 2, and if A is compatible with Σ, then Σ is q-undemanding. The rest of the paper is devoted mostly to the proofs of Theorems 1–8. After a brief development of homotopy and free groups in §§4–6, the proof of Theorem 1 is completed in §7. After a brief development of the cohomology ring in §9, the proofs of Theorems 2–5 are completed in §10. After a brief development of the Hopf invariant in §11, the proof of Theorem 7 is completed in §12. The proof of Theorem 8 is sketched in §13.
3.4
A lemma for all the proofs.
Although the topological methods vary, there is one simple lemma that unites our purposes throughout. Recall, that in all of Theorems 1–7, the hypothesis is that Σ can be modeled by continuous operations F t on a certain space A, and the conclusion is that Σ is undemanding, i.e., that Σ can be modeled by operations Ft0 on some set B, with each Ft a projection or a constant. Recall, also, that our definition of satisfaction by operations F t involves the so-called term-operations τ (and τ N ) that were are constructed (by recursion on |τ |) in Equations (3–7). Also the terms τ 0 have their recursive construction from the operations Ft0 . Lemma 9 Given operations F t defined on A for t ∈ T , and operations Ft0 defined on B for t ∈ T . If it is possible to define each term operation τ 0 : B ω −→ B directly from the term operation τ : Aω −→ A (i.e., without reference to the syntax of τ ), and if the operations F t satisfy Σ, then the operations Ft0 also satisfy Σ. The same conclusion holds if there is an algorithm defining τ 0 directly from N τ whenever N is bigger than the subscript of any variable appearing in τ . Proof. Consider an equation σ ≈ τ from Σ. Since the operations F t satisfy σ ≈ τ , the term operations σ and τ are identical (as operations defined on Aω ). It follows from our hypothesis that the term operations σ 0 and τ 0 are identical (as operations defined on B ω ). In other words, the operations Ft0 satisfy σ ≈ τ .
14
Since this was an arbitrary equation from Σ, we see in fact that the operations Ft0 satisfy Σ. Much of the work that follows, therefore, has to do with establishing the hypothesis of Lemma 9 in each of our various contexts. Each of Lemmas 37, 46 and 52 shows how to define τ 0 from τ (or from a τ ? that is readily obtained from τ ). As one might imagine, those lemmas are proved by induction on |τ |.
4
The path groupoid of a topological space.
To every topological space A there is associated an algebraic object known as its path groupoid or fundamental groupoid, denoted Π(A). In this section we will define Π(A) and present some of its properties. The fundamental group or first homotopy group of A will appear (in Lemma 10) as a subgroup of Π(A). (This definition of the path groupoid agrees with the one on page 139 of [24]. See also [40].) The elements of Π(A) are the equivalence classes of continuous maps (or paths) γ : [0, 1] −→ A with respect to the relation of homotopy with endpoints fixed. More precisely, for two maps γ, γ 0 : [0, 1] −→ A
(18)
we define γ ∼ γ 0 (γ is homotopic to γ 0 (with fixed endpoints)) to mean that there exists a continuous function H : [0, 1]2 −→ A
(19)
such that H(x, 0) H(x, 1) H(0, t) H(1, t)
= = = =
γ(x) γ 0 (x) γ(0) γ(1),
(20) (21) (22) (23)
for 0 ≤ x, t ≤ 1. Notice that that Equation (21) (with x = 0) and Equation (22) (with t = 1) together imply that γ(0) = γ 0 (0). Similarly, Equations (21) and (23) imply that γ(1) = γ 0 (1). We omit the proof that ∼ is an equivalence relation on the set of all paths. The equivalence class of γ over ∼ is denoted [γ]. Finally, Π(A) is defined to be the class of all [γ]. A binary operation (“product”) is defined on the set Π(A) as follows. If γ and δ are paths, as defined above, and if γ(1) = δ(0), then the product γ · δ : [0, 1] −→ A is defined by γ(2x) 0 ≤ x ≤ 12 γ · δ(x) = δ(2x − 1) 1 ≤ x ≤ 1. 2 15
It is not hard to check that the homotopy class of γ · δ depends only on the homotopy classes of γ and δ, and hence [γ] · [δ] can unambiguously be defined as [γ · δ]. It is not hard to prove that if β is a constant map with value equal to γ(0), and if δ is a constant map with value equal to γ(1), then β·γ γ·δ
∼ ∼
γ δ.
(24) (25)
Thus the appropriate constant maps serve as left and right units for the multiplication of homotopy classes. It is also not hard to prove that if γ −1 is defined by (26) γ −1 (x) = γ(1 − x), then γ · γ −1 γ −1 · γ
∼ ∼
γ(0) γ(1)
(27) (28)
(with the right-hand sides denoting constant maps). Finally, the reader may check that, if the endpoints match appropriately for all the products to be defined, then β · (γ · δ) ∼ (β · γ) · δ (29) for any paths β, γ and δ. For a ∈ A, we define the set of loops at a to be the subset of Π(A): Πa (A) = {[γ] ∈ Π(A) : γ(0) = a = γ(1) }.
(30)
Lemma 10 (Πa (A), ·, −1 ) is a group (whose unit element is the constant path with value a). (This group is frequently known as the fundamental group of A or the first homotopy group of A.) Proof.
The result is immediate from equations (29), (24), (25), (27) and (28).
The reader can easily prove that if there is a path from a to b in A, then Πa (A) ∼ = Πb (A). See also Lemma 15 below. Thus in path-connected spaces, all the fundamental groups are isomorphic. At the abstract level, a category is a set with a partial operation that satisfies Equation (29), and which for all γ has β and δ satisfying Equations (24) and (25). If for each γ there exists γ −1 satisfying Equations (27) and (28), then the category is called a groupoid. The construction of Πa (A) from Π(A) has a counterpart in category theory of forming the monoid of self-maps of a given object. If the category is a groupoid, then the individual monoid is a group. Since they are not necessary for our work in this paper, we omit the precise 16
form of these abstract statements. The reader may consult any basic work on category theory, or, for example, §3.6 of [24]. Also see Higgins [16], [6] and [7]. With no further assumptions, Π(A) might consist only of constant paths, or indeed it might be the case that there are many non-constant paths, but that any two paths with the same endpoints are homotopic to each other. (The first possibility occurs for a totally disconnected space like the Cantor set or the rational line; the second occurs e.g. for Euclidean space Rn .) In such extreme cases Π(A) contains no useful information. The fortunate fact is that some spaces A — such as A a figure-eight — have highly complex and non-trivial Π(A). In fact this A has Πa (A) a free group on two generators (regardless of the choice of a).
5
The path groupoid of a topological algebra
Now suppose that the space A of §4 is equipped with some continuous operations Ft for t ∈ T , in other words, that (A, Ft )t∈T is a topological algebra based on A. We first observe that paths can be subjected to the operations Ft , simply by performing the operations pointwise. In other words, we extend the operations Ft to paths γi : [0, 1] −→ A as follows: Ft (γ1 , . . . , γn(t) )(x) = Ft (γ1 (x), . . . , γn(t) (x))
(31)
for t ∈ T and 0 ≤ x ≤ 1. In order to define counterparts of Ft on Π(A), we next need to consider the homotopy relation. Lemma 11 The homotopy relation ∼ is a congruence relation on the algebra of all paths. In other words, If γi ∼ δi for 1 ≤ i < n(t), then Ft (γ1 , . . . , γn(t) ) ∼ Ft (δ1 , . . . , δn(t) ) Proof.
We are given functions Hi : [0, 1]2 −→ A such that Hi (x, 0) Hi (x, 1) Hi (0, t) Hi (1, t)
= = = =
γi (x) δi (x) γi (0) γi (1)
for 1 ≤ i ≤ n(t). It is not hard to check that H(x, t) = Ft (H1 (x, t), . . . , Hn(t) (x, t)) brings about the homotopy relation that is asserted in the lemma.
17
Now, in the usual way, one can form the quotient algebra with respect to homotopy of the algebra of paths under the operations Ft (for t ∈ T ). By §4 its universe is Π(A), and so we have constructed an algebra (Π(A), Ft? )t∈T whose operations
Ft∗ : Π(A)n(t) −→ Π(A)
are defined via Ft? ([γ1 ], . . . , [γn(t) ]) = [Ft (γ1 , . . . , γn(t) )]. Lemma 12 Let Σ be a set of equations in the operation symbols Ft (t ∈ T ). If continuous operations Ft model Σ on A, then the operations Ft? model Σ on Π(A). In other words If
(A, Ft )t∈T |= Σ,
then
(Π(A), Ft? )t∈T |= Σ
Proof. Obvious from the fact that Π(A) is a quotient of the algebra of paths, which itself is a subalgebra of the power (A, Ft )[0,1] . Lemmas 11 and 12 do not in themselves imply any particular advantage to the path-algebra Π(A). The real advantage of this algebra comes in the combination of Lemma 12 with the following lemma. Lemma 13 The operations Ft? (t ∈ T ) commute with the multiplication of paths in Π(A). In other words, if [γi ], [δi ] ∈ Π(A) (1 ≤ i ≤ n(t)), and if [γi ] · [δi ] is defined for each i, then Ft? ([γ1 ] · [δ1 ], · · · , [γn(t) ] · [δn(t) ]) = Ft? ([γ1 ], · · · , [γn(t) ]) · Ft? ([δ1 ], · · · , [δn(t) ]). (32) Proof.
One easily verifies, from the definitions involved, that Ft (γ1 · δ1 , · · · , γn(t) · δn(t) )
and Ft (γ1 , · · · , γn(t) ) · Ft (δ1 , · · · , δn(t) ) define the same path. Their homotopy equivalence classes are the two sides of Equation (32), and hence the lemma is proved. Lemma 13 may be summarized by saying that each operation Ft? is a groupoid-homomorphism. We mostly use the following specialization of Lemma 13 to the case where the γi and δi are loops at a single point a. The full pathgroupoid is useful as a context for establishing satisfaction of Σ (as we did in Lemma 12); on the other hand, for calculations about homomorphisms, it is more useful to work at the level of Πa (A), since we have a whole theory of group homomorphisms to draw on (see §6 below). 18
Lemma 14 For each t ∈ T and each a ∈ A, the operation Ft? maps the group Πa (A) to the group ΠF (a,···,a) (A). Moreover the resulting map, which we also denote Ft? : Πa (A)n(t) −→ ΠFt (a,...,a) (A), is a homomorphism of groups. Proof.
Immediate from Lemma 13.
Remark. Lemma 14 interests us especially in the case where the fundamental groups Πa (A) are free on two (or more) generators, since, as we shall see in §6, the homomorphisms between such groups are few and limited. Lemma 15 If A is pathwise connected, the homomorphism in Lemma 14 is independent, up to isomorphisms, of the choice of a. More precisely, for all a, b ∈ A there are group isomorphisms λ : Πa (A) µ : ΠFt (a,···,a) (A)
−→ Πb (A) −→ ΠFt (b,···,b) (A)
such that, Ft? (λ([γ1 ]), · · · , λ([γn(t) ])) = µ(Ft? ([γ1 ], · · · , [γn(t) ]))
(33)
for all [γ1 ], . . . , [γn(t) ] ∈ Πa (A). Proof. Let β : [0, 1] −→ A be a path with β[0] = a and β[1] = b. Let ρ be the path from Ft (a, · · · , a) to Ft (b, · · · , b) defined by ρ(x) = Ft (β(x), · · · , β(x)). Now define λ([γ]) µ([δ])
[β −1 · γ · β] [ρ−1 · δ · ρ]
= =
for γ ∈ Πa (A) and δ ∈ ΠFt (a,···,a) (A). The verification of Equation (33) follows directly from Lemma 13.
6
Free Groups.
In this section we consider maps from a finite power of a finitely generated free group, to another finitely generated free group. We assume that the reader has some background in the general subject of free groups. (See for instance Hall 19
[15] or Magnus, Karrass and Solitar [25]. There is also a short development of free groups on pages 119–120 of [24].) The only thing one needs to carry from §6 to the rest of the paper is Lemma 24, which rather strictly curtails the homomorphisms that are possible from a power Gn of a free group to G. The application of Lemma 24 (in §7) will be to the group homomorphisms described in Lemma 14 above. Lemmas 16–19 are well known, and so we merely sketch the arguments. (For example, Lemmas 17 and 19 appear as Problems 1.4.2 and 1.4.4 on page 41 of [25].) Lemma 16 If any cancellation occurs during the formation a power wn of a reduced word w, then the first letter of wn cancels the last letter of wn . Proof. Consider how cancellation works in forming the power wn . The first literal power cancels the last, the next one at the beginning cancels the next one back from the end, and so on. If this process should reach the middle of the word, then the entire word would self-cancel (beginning from the middle outward), and we would have w = 1. No cancellation occurs in the formation of powers of 1. Lemma 17 If word w is cyclically reduced, i.e. ends in a different letter than it begins, or if it is a power of a generator, and n 6= 0, then wn = v n
implies
w = v.
Proof. Obviously no cancellation occurs in the formation of wn , and hence it begins and ends in different letters. Therefore v n also begins and ends in different letters (since v n = wn ). Hence, by Lemma 16, no cancellation occurs in the formation of v n . Now the conclusion is obvious. The other case is w = am for a some letter. Hence v n = wn = amn . According to Lemma 16, no cancellation occurs in the formation of v n . Again, the conclusion is obvious. Lemma 18 If w is any word, and n 6= 0, then wn = v n
implies
w = v.
Proof. This is immediate from Lemma 17 and the fact that for every word w there is a word s such that s−1 ws is either cyclically reduced or a power of a single letter. Lemma 19 If w and v are any words, and n 6= 0, then v wn = wn v
implies 20
wv = vw.
Proof. The hypothesis implies that wn = (v −1 wv)n , and hence, by Lemma 18, that w = v −1 wv. Thus vw = wv. Lemma 20 Suppose that w1 , w2 , . . . , wn are elements of a free group, none of them equal to the unit element 1, and that they commute in pairs down the list. (I.e., w1 w2 = w2 w1 , w2 w3 = w3 w2 , and so on.) Then there is a cyclic subgroup Z that contains all the wi . Proof. By induction on n. If n = 1, the conclusion is obvious. For the inductive step, we let n > 1, and suppose that there is a cyclic subgroup Z0 containing w1 . . . wn−1 . In other words, there exists x and integers ni 6= 0 such that wi = xni for i = 1, 2, . . . , n − 1. We are given that wn commutes with wn−1 , which is a non-zero power of x. Hence, by Lemma 19, x commutes with wn . The subgroup Z generated by x and wn must therefore be commutative, and moreover it is free, since every subgroup of a free group is free. The only commutative free group is the free group on one generator, i.e., a cyclic group. Thus Z is a cyclic group that contains all the wi . Lemma 21 If F : Gn −→ H, where G is a finitely generated group, and H is a free group, then either (i) F [Gn ] is contained in a one-generated free subgroup of H or (ii) F depends on only one variable, Proof. Let G be generated by {gj : j = 1, . . . , m}. For 1 ≤ i ≤ n and 1 ≤ j ≤ m, define gij hij
= =
(1, · · · , 1, gj , 1, · · · , 1) ∈ Gn F (gij ) ∈ H,
where the gj appears in the ith place of gij . Clearly gij · grs = grs · gij for i 6= r, since all the component products have 1 as at least one of their factors. Since F is a homomorphism, we also have hij · hrs = hrs · hij
for i 6= r.
The proof now revolves around the set K = { (i, j) : hij 6= 1 }. The reader should check that either Case 1 or Case 2 must hold for K. 21
(34)
Case 1: K contains pairs (i1 , j1 ) and (i2 , j2 ) with i1 6= i2 . It is not hard to check that K can be enumerated as a finite sequence (possibly with repetitions), (i1 , j1 ), (i2 , j2 ), . . . , (iN , jN ), such that ik 6= ik+1 for k = 1, 2, . . . , N − 1. Thus, by (34), the sequence wk = hik jk satisfies the hypotheses of Lemma 20: none of the wk is 1, and wk commutes with wk+1 for k = 1, 2, . . . , N − 1. Therefore all the wk lie in a commutative subgroup of H. Since the wk are easily seen to generate the image of F , it follows that F [Gn ] is contained in a commutative subgroup of the free group H. A commutative subgroup of a free group is free on one generator; hence we have proved that alternative (i) of the lemma holds in Case 1. Case 2: For some i K ⊆ { (i, j) : j = 1, . . . , m }.
(35)
We will prove that F (x1 , . . . , xn ) depends only on xi . Referring to the definition of K and the definition of the elements hij , we easily see the following translation of (35): if gj is one of the generators of G, then (1, · · · , 1, gj , 1, · · · , 1) ∈ ker(F )
(36)
(where gj is placed anywhere except in the ith component). Clearly the vectors appearing in (36) generate all vectors that have 1 in the ith place, and so we have (x1 , · · · , xi−1 , 1, xi+1 , · · · , xn ) ∈ ker(F ), (37) for any elements xk of G (1 ≤ k ≤ n; k 6= i). Now to prove that F depends only on xi , we consider vectors x = (x1 , · · · , xn ) and y = (y1 , · · · , yn ) in Gn with xi = yi . We need to show that F (x) = F (y). By (37) we have −1 −1 xy −1 = (x1 y1−1 , · · · , xi−1 yi−1 , 1, xi+1 y1+1 , · · · , xn yn−1 ) ∈ ker(F ).
Hence F (x) = F (xy −1 ) · F (y) = 1 · F (y) = F (y). Hence we have proved that alternative (ii) of the lemma holds in Case 2. Our next lemma is actually a rather deep theorem in free-group theory. We will not include a proof. We quote Hall’s version of the statement. Lemma 22 A free group Fr with a finite number r of generators is freely generated by any set of r elements which generate it. Proof. See Theorem 7.3.3 on page 109 of Hall [15], or Corollary 2.13.1 on page 110 of Magnus, Karrass and Solitar [25].
22
Lemma 23 If G is a free group on k generators (for some k < ω), and f : G −→ G maps onto G, then f is one-to-one. Proof. Let α1 , . . . , αk be free generators of G. Since f is onto, f (α1 ), . . . , f (αk ) generate G; by Lemma 22, they freely generate G. Hence there exists a homomorphism h : G −→ G such that h(f (αj )) = αj
(1 ≤ j ≤ k)
Since the αi generate G, h◦f is the identity map, and hence f is one-to-one. Lemma 24 If G is a free group on k generators (for some k with 2 ≤ k < ω), and if F : Gn −→ G maps onto G, then there exists i (1 ≤ i ≤ n) and there exists an automorphism φ of G such that F (x1 , · · · , xn ) = φ(xi ) for all x1 , . . . , xn ∈ G. Proof.
7
Immediate from Lemmas 21 and 23.
The proof of Theorem 1.
As in the statement of the theorem, we let A be a path-connected topological space whose fundamental group is free on k generators (2 ≤ k < ω) — for instance, a figure-eight space. We assume that A is compatible with a set Σ of equations. That is, we are given continuous operations Ft : An(t) −→ A such that (A, Ft )t∈T |= Σ. Our objective is to prove that Σ is undemanding. In other words, we need to find special operationS Ft0 modeling Σ, i.e. operations Ft0 on a set B with more than one element, such that (B, Ft0 )t∈T |= Σ and such that each Ft0 is either a constant operation or a projection operation. According to §4 and §5, we have operations Ft? defined on the groupoid Π(A) such that (i) (Π(A), Ft? )t∈T |= Σ (Lemma 12). (ii) The subset Πa (A) has the structure of a free group on k generators (Lemma 10). 23
(iii) Restricted to Πa (A), each operation Ft? is a group-homomorphism from Πa (A)n(t) to ΠF (a,···,a) (A) (Lemma 14). (iv) The homomorphism of (iii) is independent of a (Lemma 15). Definition of the operations Ft0 : We now define the set B and the operations Ft0 on B. 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 Ft? is not onto (this condition is independent of a, by (iv)), we define Ft0 (x1 , · · · , xn(t) ) = c
(B) If Ft? is onto, then by Lemma 24 we have Ft? (x1 , · · · , xn(t) ) = φ(xi ) for some i and some automorphism φ. In this case we define Ft0 (x1 , · · · , xn(t) ) = xi This completes the definition of the operations Ft0 for t ∈ T . Evidently each Ft0 is either a constant or a projection operation. What remains is to show that they model Σ. In Equations (3) and (4) we saw how an interpretation of symbols Ft by operations Ft leads to an interpretation of any term τ by a function τ : Aω −→ A. It is merely a change of notation to do the same thing for the interpretations Ft? : they lead in the same way to an associated interpretation τ ? : Gω −→ G. Similarly, interpretations Ft0 of the operation symbols lead to an interpretation τ 0 of each term τ . In these terms, our plan for the rest of §7 can be expressed as follows: we are given that σ? = τ ? for an equation σ ≈ τ of Σ; we need to prove that σ0 = τ 0 . It should be noted, in particular, that when restricted to Πa (A), the operation τ ? maps into Πτ (a,···,a) . When we say that τ ? is onto we mean that it has Πτ (a,···,a) as its image when restricted to Πa (A). Again by (iv), this condition is independent of a. Lemma 24 and (B) are special cases of the next Lemma. Lemma 25 If τ ? is onto, then τ ? (x) = φ(xj ) for some j and some automorphism φ of the fundamental group. In this case τ 0 (x) = xj Proof. The proof is by induction on |τ |. If τ is a variable, the conclusion clearly holds (with φ taken as the identity map). Otherwise, by Equations (3–5), τ is formed as Ft (τ1 , · · · , τn(t) ), and ? τ ? (x) = Ft? (τ1? (x), · · · , τn(t) (x)).
24
(38)
Since τ ? was assumed to be onto, we know that Ft? must also be onto. Hence, by Lemma 24, Ft? (x1 , · · · , xn(t) ) = ψ(xi ) (39) for some i and some automorphism ψ. By (B), we have Ft0 (x1 , · · · , xn(t) ) = xi for all x. It follows immediately that τ 0 (x) = τi0 (x).
(40)
Now from Equations (38) and (39) we immediately deduce that τi? = ψ −1 ◦τ ? and hence that τi? is onto. Therefore, by induction, τi? (x) = λ(xj )
(41)
for some j and some automorphism λ, and moreover τi0 (x) = xj .
(42)
Now by Equations (38), (39) and (41), we have τ ? (x)
= Ft? (τ1? (x), · · · , τn? (x)) = ψ(τi? (x)) = ψ(λ(xj )) = φ(xj )
(where φ = ψ◦λ). And by Equations (40) and (42), τ 0 (x) = xj . Lemma 26 If τ ? is not onto, then τ 0 (x) = c. Proof.
The proof is by induction on |τ |. Clearly τ is not a variable, so τ = Ft (τ1 , · · · , τn(t) )
(43)
for some terms τ1 , . . . , τn(t) . Case 1: Ft? is not onto. Then Ft0 (x1 , · · · , xn(t) ) = c, by (A). Clearly then τ 0 is the same constant, and the proof is complete in this case. Case 2: Ft? is onto. In this case, by (B), Ft? (x1 , · · · , xn(t) ) = φ(xi ) for some i and some automorphism φ, and moreover Ft0 (x1 , · · · , xn(t) ) = xi . 25
(44)
From Equations (43) and (44) we have ? τ ? (x) = Ft? (τ1? (x), · · · , τn(t) (x)) = φ(τi? (x))
and so τi? is not onto. By induction, τi0 (x) = c, and so τ 0 (x) = Ft0 (τ10 (x), · · · , τn0 (x)) = τi0 (x) = c.
Completion of the proof of Theorem 1. We begin by establishing the hypothesis of Lemma 9 (from §3). Clearly Lemma 12 implies that the operation τ ? depends only on the operation τ , and clearly Lemmas 25 and 26 define τ 0 from the operation τ ? . All in all, we have τ 0 defined from the operation τ , and so the hypothesis of Lemma 9 is satisfied. Thus the operations Ft0 satisfy Σ, and hence Σ is undemanding. This completes the proof of Theorem 1.
8
Commutative graded rings; tensor products.
In §9 below, we will summarize the needed facts about the (absolute) cohomology ring H ? (A, R), with coefficients in a fixed commutative ring R with unit. Here we preface that section with a discussion of commutative graded rings over R. In some contexts, e.g. the proof of Lemmas 27 and 36 below, we will need R to be an integral domain R, i.e. a commutative ring with unit satisfying If
λµ = 0,
then
λ = 0
or
µ = 0.
(45)
Most of our proofs rely on Theorem 32 below, which assumes that R is a field. In some of our work, such as Lemma 39 of §26, it will be necessary to assume that the characteristic of R is not 2. On the other hand, in §10.4 (the Klein bottle) and §10.5 (real projective space), we will work with the field Z/2 of integers modulo 2. Then, in defining degrees in §11 below, we will take R to be Z, the ring of integers. Lemma 55 below also requires integral coefficients.
8.1
The essential facts.
A graded ring over R is an associative bilinear algebra H over R with unit (see page 15 of [24]), which has designated R-submodules Hi (i ∈ ω) such that M (i) H = Hi i∈ω
(ii) Hi Hj ⊆ Hi+j . It follows, of course, that the unit element 1 lies in H0 . A commutative graded ring over R (to which we will refer as an R-CGR) is a graded ring over R that also satisfies 26
(iii) xy = (−1)pq yx for x ∈ Hp and y ∈ Hq . A homomorphism from an R-CGR H to an R-CGR K is a homomorphism f : H −→ K of bilinear algebras that also satisfies (i) f [Hp ] ⊆ Kp (ii) f (hk) = (−1)pq f (h)f (k), for p, q ∈ ω, h ∈ Hp and k ∈ Hq . The tensor product H ⊗ K of R-CGRs H and K is the R-CGR with the following presentation. Its generators are all ordered pairs (h, k) ∈ H × K. Such a pair, in the context of the tensor product, is traditionally denoted h ⊗ k. The relators for the presentation are (i) all relations of R-multilinearity: r(h ⊗ k) = (rh) ⊗ k = h ⊗ (rk) (h1 + h2 ) ⊗ k = (h1 ⊗ k) + (h2 ⊗ k) h ⊗ (k1 + k2 ) = (h ⊗ k1 ) + (h ⊗ k2 ) (ii) (h1 ⊗ k1 ) · (h2 ⊗ k2 ) = (−1)pq (h1 h2 ⊗ k1 k2 ). for8 h2 ∈ Hp and k1 ∈ Hq . Finally, (iii) H ⊗ K is made into a graded algebra by defining (H ⊗ K)p = { h ⊗ k : (∃s ≤ p)(h ∈ Hs and k ∈ Kp−s ) } The reader may easily check that the mapping η1 : h 7−→ h ⊗ 1
(46)
is a homomorphism η1 : H −→ H ⊗ K — called the first-coordinate injection — and that η2 : k 7−→ 1 ⊗ k (47) is a homomorphism η2 : H −→ H ⊗ K — second-coordinate injection. Moreover, categorically speaking, the diagram 8 The reader who chooses to do so may disregard the minus signs that crop up for elements of odd degree, both here and in the foregoing definitions of commutativity and homomorphisms. In fact, our main example has H2n+1 = {0} for n ≥ 0. Nevertheless, the minus sign explains why we must say “of even degree” in Theorem 2. In §10.4 and §10.5, the minus sign will not appear; the characteristic is 2.
27
PPηP PPq H ⊗ K 1 η K H
1
2
is a co-product. In other words, given R-CGR homomorphisms f1 : H −→ G, f2 : K −→ G, there is a unique homomorphism f : H ⊗ K −→ G such that f ◦ηi = fi (for i = 1, 2). In other words, the diagram f H PPη PPPq ) f G 1 H ⊗ K iPPP P f PK η 1
1
2
2
commutes. (The interested reader may prove this for himself.) We close §8 with §8.2, which contains some rather particular results on RCGR’s. They will be useful in the proofs of Theorems 3, 4 and 6 in §10 below. (Lemma 28 appears in the proofs of Lemma 45 and Lemma 41, and Lemma 30 appears in the proof of Lemma 39.) On a first reading one may be well advised to skip §8.2 and proceed directly to §9. Then one could focus first on the proof of Theorem 2, which epitomizes the cohomological method, without requiring the technicalities of §8.2.
8.2
Some technical lemmas on CGR’s.
Lemma 27 Suppose that H is an R-CGR, and that a, c ∈ Hp , with a 6= 0. Then there exists λ ∈ R such that, for all b, d ∈ Hq , if a ⊗ b = c ⊗ d,
(48)
then b = λd. Moreover, if either b 6= 0 or d 6= 0, then Equation (48) implies also that c = λa. Proof. Let us first assume Equation (48) with a 6= 0. Let h1 , . . . , hm be a basis of the vector space Hp and let k1 , . . . , kn be a basis of the vector space Hq . We then express a, b, c, d as a =
c =
m X i=1 m X
αi hi
b =
γi hi
d =
i=1
n X j=1 n X j=1
28
βj kj δj kj
It now follows from the R-multilinearity relations (i) on page 27 that a⊗b = c⊗d =
m X n X
αi βj hi ⊗ kj
i=1 j=1 m X n X
γi δj hi ⊗ kj
i=1 j=1
From the well-known fact that hi ⊗ kj (1 ≤ i ≤ m, 1 ≤ j ≤ n) are linearly independent elements of H ⊗ H, we see immediately that αi βj = γi δj
(49)
for 1 ≤ i ≤ m, 1 ≤ j ≤ n. Now since a 6= 0, we have αr 6= 0 for some r. Thus taking r for i in Equation (49), we have βj =
γr δj αr
for 1 ≤ j ≤ n. In other words, for λ = γr /αr , we have b = λ d, and the first clause of the lemma is established. Suppose now that b 6= 0. Then clearly βs 6= 0 for some s, and from Equation (49) we have αr βs = γr δs . (50) The left-hand side is the product of two non-zero elements, hence non-zero (in an integral domain). Therefore δs 6= 0. On the other hand, if we assume that d 6= 0, then we also (trivially) obtain δs 6= 0 for some s. Thus, for the rest of the proof, we will simply assume that δs 6= 0. From Equation (50) we obtain γr βs = = λ. δs αr Now one more application of Equation (49) yields γi =
βs αi = λαi δs
for 1 ≤ i ≤ m, and hence c = λa. For an R-CGR H, we will call Hp a prime homogeneous component of H if p = m1 + · · · + mn (with n > 1 and each mi > 0) implies that Hmi = {0} for some i. For example, H1 is always a prime homogeneous component.
29
Lemma 28 Suppose that p is odd, and that Hp is a prime homogeneous Ncomn ponent of an R-CGR H. Suppose that z , z lie in this component of H, 1 2 Nn i.e., z1 , z2 ∈ ( H)p . If z1 z2 = 0, then either (a) the space generated by z1 and z2 is one-dimensional, or (b) there exists i, with 1 ≤ i ≤ n, such that zk = 1 ⊗ · · · ⊗ 1 ⊗ aik ⊗ 1 ⊗ · · · ⊗ 1
(51)
for k = 1, 2. Proof. It follows immediately from prime homogeneity that every element of Nn ( H)p is a sum of elements of the form 1 ⊗ · · · ⊗ 1 ⊗ w ⊗ 1 ⊗ · · · ⊗ 1.
(52)
In particular, we have zk =
n X
1 ⊗ · · · ⊗ 1 ⊗ aik ⊗ 1 ⊗ · · · ⊗ 1,
(53)
i=1
where aik appears in the ith place, and where k = 1, 2. If z1 = 0 or z2 = 0, then (a) holds, and we are done. Otherwise, ar1 6= 0 for some r, and as2 6= 0 for some s. If r = s and r and s are both unique, then (b) holds, and we are done. Otherwise, either r or s can be changed, and so we have ar1 6= 0
and
as2 6= 0,
(54)
for some r and s with r 6= s. Using distributivity and Equation (ii) on page 27 (remembering that p is odd), we obtain 0 = z1 z2
=
W + −
n X
1 · · · 1 ⊗ ai1 ⊗ 1 · · · 1 ⊗ aj2 ⊗ 1 · · · 1
ij
where ai1 appears in the ith place and aj2 appears in the j th place, and where W is a sum of terms of the form (52) with w ∈ H2p . Re-indexing the second sum, we obtain 0 = z1 z2
=
W +
n X
1 · · · 1 ⊗ ai1 ⊗ 1 · · · 1 ⊗ aj2 ⊗ 1 · · · 1
i
