CLASSICAL LINK INVARIANTS AND

THE HAWAIIAN EARRINGS SPACE

Tim D. Cochran

Mathematics Department, Rice University P.O. Box 1892, Houston, Texas 77251–1892

Abstract. We show that, in search of link invariants more discriminating than Milnor’s \overline{μ}-invariants, one is naturally led to consider seemingly pathological objects such as links with an infinite number of components and the join of an infinite number of circles (Hawaiian earrings space). We define an infinite homology boundary link, and show that any finite sublink of an infinite homology boundary link has vanishing Milnor’s invariants. Moreover, all links known to have vanishing Milnor’s invariants are finite sublinks of infinite homology boundary links. We show that the exterior of an infinite homology boundary link admits a map to the Hawaiian earrings space, and that this may be employed to get a factorization of K.E. Orr’s omega-invariant through a rather simple space.

Keywords: link, Milnor’s invariants, Massey products, boundary link, homology boundary link, concordance

Chapter 1.

This research was motivated by the desire to classify links of circles in S^{3} (or R^{3}) up to concordance. Recall that a link L = {K_{1}, …, K_{m}} of m-components is an ordered collection of disjointly embedded, oriented, smooth submanifolds of S^{3} where each K_{i} is homeomorphic to S^{1}. The links L_{0} and L_{1} are concordant if there exists a disjoint collection

of oriented annuli ∐_{i=1}^{m} S^{1} ×[0, 1] embedded as a smooth submanifold of S^{3} ×[0, 1] such that

the image of ∐_{i=1}^{m} S^{1} × {0} is L_{0} and ∐_{i=1}^{m} S^{1} × {1} is L_{1}. The most powerful invariants

of link concordance which do not depend strongly on the knot type of the individual components (and hence are more purely invariants of linking) are Milnor’s \overline{μ}-invariants or, equivalently, Massey products in the link exterior ([16] [12], see [20] for a precise statement of the equivalence, [13]). These have also been related to linking numbers among derived

Typeset by AMS-TEX

links [4]. The driving question in this field has been and is, “if Milnor’s \overline{μ}-invariants vanish for L, what can be said of the concordance class of L?” Milnor’s invariants are now known to vanish for boundary links and more generally for sublinks of homology boundary links [12] [4, Thm. 8.10] [7] [5]. Recall that L is a boundary link if there exists a collection {V_{1}, …, V_{m}} of compact, oriented surfaces (called Seifert surfaces) disjointly embedded in the link exterior E(L) ≡ \overline{S^{3} − (L × D^{2})}, such that ∂V_{i} is K_{i} × \{∗\} ⊆ L × ∂D^{2} (a longitude of K_{i}). An homology boundary link is defined similarly except that ∂V_{i} is allowed to be any disjoint union of longitudes of components of K such that [∂V_{i}] in H_{1}(∂E(L)) is equal to [K_{i} × {∗}], that is to say that ∂V_{i} is merely “algebraically” equal to a longitude of K_{i}. A construction of Pontryagin yields equivalent definitions which shall be crucial to the theme of our paper. By choosing tubular neighborhoods of the V_{i}, one obtains a map from E(L) to ⋁_{i=1}^{m} S^{1} by sending each [−1, 1] fiber of the i^{th} tubular neighborhood around the i^{th}

circle and all other points to the wedge point, In fact, L is a boundary link (respectively

homology boundary link) if and only if there is a continuous map f : E(L) → ⋁_{i=1}^{m} S^{1} which

sends the i^{th} meridian to the i^{th} circle (respectively to a loop homologous to the i^{th} circle) [21].

In the present paper, we will show that, in search of invariants more discriminating than Massey products, and in search of new classes of links with vanishing Massey products, one is naturally led to consider infinite analogues of the above objects which might in other contexts be seen as “pathological”. In particular we define an homology boundary link with an infinite number of components and show that any finite sublink of an infinite homology boundary link has vanishing Milnor’s invariants (Thm. 2.6). Conversely, we show that a link which satisfies a geometric condition which seems to be (but may not be) equivalent to the vanishing of the \overline{μ}-invariants, is indeed a sublink of an infinite homology boundary link.

Building on this work, Kent Orr has announced such a converse which assumed only the vanishing of the \overline{μ}-invariants. However his definition of an infinite homology boundary link seems to be weaker than ours. Since this work has not yet appeared, the status of a complete converse is, in any case, in doubt.

We propose that this class, sublinks of infinite homology boundary links, is interesting and potentially new. Unfortunately, as of this writing, we do not know if this class is strictly larger than the class of sublinks of finite homology boundary links!

Since the present work, the author and Kent Orr have shown that most sublinks of (finite) homology boundary links are not concordant to any boundary link [8, 9]. Therefore the pressing question has become “If Milnor’s invariants vanish for L, is L concordant to a sublink of an homology boundary link (or even an homology boundary link)?” Indeed it may be that \overline{μ}(L) = 0 if and only if L is a sublink of an homology boundary link! Consequently our proposed new class of links becomes even more relevant.

Moreover, to complete the analogy of the infinite case to the finite case, we show that if L_{∞} is an infinite homology boundary link then there is an induced map from E(L_{∞}) to the countably infinite wedge of circles with the “Hawaiian Earrings” topology, i.e., to H = {(x, y) ∈ R^{2} | (x − 2^{−n})^{2} + y^{2} = (2^{−n})^{2}n ∈ Z^{+} ∪ {0}} in the subspace topology.

Finally, we will show that this line of attack may shed some light on other existing invariants. In particular, Kent Orr has defined, for any based link (L, b) which has vanishing \overline{μ}-invariants, an element θ_{ω}(L, b) ∈ π_{3}(\overline{K}_{ω}) where \overline{K}_{ω} is the cofiber of the map of classifying spaces K(F, 1) → K(F^{∧}, 1) induced by the natural embedding F → F^{∧} of the free group on m-letters in its nilpotent completion [18, 19]. This is invariant under a “based concordance”. It is of great interest therefore to decide whether or not \overline{μ} = 0 implies θ_{ω} = 0. Activity as regards this important question has lanquished due to the seeming difficulty of analyzing π_{3}(\overline{K}_{ω}) and to the total lack of possible counter-examples (sublinks of homology boundary links also have vanishing θ_{ω} [5; Corollary 2.5]) (however see [14] for important progress). The class of sublinks of infinite homology boundary links may provide such counter-examples. We shall show that if L is a sublink of an infinite homology boundary link L_{∞}, then θ_{ω}(L) : S^{3} → \overline{K}_{ω} factors through a “simpler” space which contains H and which is the inverse limit of finite 2-complexes each of which is homotopy-equivalent to a wedge of 2-spheres. This space is essentially combinatorially defined from H using the data of which longitudes ∂V_{i} contains (compare the “pattern” of [7]).

It is our point of view that these spaces should be viewed as constituting part of the 2skeleton of the mysterious space \overline{K}_{ω}. If this class of spaces could be shown to have π_{3} = 0 then θ_{ω} would vanish for all sublinks of infinite homology boundary links. On the other hand, if even one of these “simpler” spaces is (constructively) shown to have non-zero π_{3} then this might lead to its realization by a link L, shedding light on the possible gap between θ_{ω} and the \overline{μ}-invariants. We remark that it is known that π_{2}(\overline{K}_{ω}) is uncountable making it likely that π_{3}(\overline{K}_{ω}) is uncountable but that many elements are not realizable as θ_{ω} of a link [2, Prop. 4.4].

Chapter 2.

Definition 2.1. A link L of J-components in S^{3} is an indexed collection (L) such that:

a) for each α ∈ J, L_{α} is a pair (L_{α}, ϕ_{α}) where ϕ_{α} is a smooth orientation-preserving embedding of S^{1} × D^{2} into S^{3}, ϕ_{α}(S^{1} × {0}) = K_{α} (with orientation), ϕ_{α}(S^{1} × {1}) = l_{α} an oriented longitude of the knot K_{α}; and

b) if α ≠ β im ϕ_{α} ∩ im ϕ_{β} is empty.

We also allow the degenerate case that K_{α} is empty. The exterior E(L) of the link L is the complement in S^{3} of ⋃_{α} ϕ_{α}(S^{1} × ^{2}). Then E(L) is a compact metric space but is

not a manifold at cluster points of ⋃_{α} ϕ_{α}(S^{1} × S^{1}).

Definition 2.2. An infinite homology boundary link L_{∞} is a link {K_{n}} indexed by the positive integers, together with a collection V = {V_{n} | n ∈ Z_{+}} of connected, oriented, compact 2-dimensional submanifolds of S^{3} with tubular neighborhoods Ψ_{n} : V_{n} × [0, 1] ↪ S^{3} whose images lie in E(L_{∞}) and are pairwise disjoint, such that:

1) Ψ_{n}(∂V_{n} × [0, 1]) ⊂ ∂E(L_{∞})

2) ∂V_{i} is a finite union of longitudes of the {K_{n}},

3) V_{i} is a generalized Seifert surface for K_{i}, i = 1, …, m that is, ∂V (x_{i}) is a union of longitudes which is homologous in H_{1}(∂E(L)) to a longitude of K_{i}.

We are obliged to convince the reader that links with an infinite number of components, in particular infinite homology boundary links, arise quite naturally from consideration of Massey products in the exterior of ordinary (finite) links. We assume the reader is acquainted with Massey products [13, 4]. Recall that the vanishing of all Massey products for the space E(L) guarantees the existence of an infinite indexed collection of 1-cochains whose coboundaries are related, in a specific inductive fashion, to their cup-products. By Poincaré duality, there ought then to exist an infinite collection of surfaces whose boundaries are related, in an inductive fashion, to their intersections. As a trivial example, if the linking number lk(K_{1}, K_{2}) = 0 then there exist Seifert surfaces V_{1}, V_{2} for K_{1}, K_{2} in E(K_{1}∪K_{2}). Then V_{1}∩V_{2} is a framed one manifold called say K_{(1,2)} and the length 4 Massey products vanish iff lk (K_{(1,2)}, K_{(1,2)}^{+}) = 0 which is to say K_{(1,2)} = ∂V_{(1,2)} where V_{(1,2)} lies in E (K_{1} ∪ K_{2} ∪ K_{(1,2)}). In [4] the philosophy was espoused that, if these surface systems were assumed to be “nicely” embedded then much intuition could be gained. It was shown that existence of such a system of surfaces implied the vanishing of the Massey products (but is a priori stronger because of the conditions imposed on the surface systems).

We formalize the latter assumptions below and retain the nomenclature Technical Condition (TC) which was introduced in [6; p. 130]. Here L = {K_{1}, …, K_{m}}. A 1 bracket is one of the letters {x_{1}, …, x_{m}}. A n -bracket is a symbol [b_{1}, b_{2}] where b_{1} is an i-bracket and b_{2} is a j-bracket with i+j = n. These brackets are used to index the infinite collection of surfaces in E(L).

Technical Condition on L (TC). There is a set V of compact, connected, oriented (possibly empty) surfaces in E(L), indexed by the set of brackets of positive weight such that:

(1) V (x_{i}) is a generalized Seifert surface for K_{i}.

(2) V (b_{1}) and V (b_{2}) are distinct and meet transversely unless they coincide modulo orientation.

(3) ∂V ([b_{1}, b_{2}]) is V (b_{1}) ∩ V (b_{2}) which is connected and oriented according to a convention not relevant to our present discussions. If V (b_{1}) and V (b_{2}) coincide, we understand that their intersection is empty.

(4) V (b_{1}) ∩ V (b_{2}) ∩ V (b_{3}) is empty unless, for some permutation (i, j, k) of (1, 2, 3) ∂V (b_{k}) is a component of the intersection of V (b_{i}) and V (b_{j}).

For our current study we must also impose a further condition 5), that stipulates that distinct surfaces have tubular neighborhoods which are disjoint except where they meet transversely. This implies, for example, that the subspace of all circles {∂V (b)} has the topology of a disjoint union of circles.

Theorem 2.3. If L satisfies TC above then L is a sublink of an infinite homology boundary link L_{∞}.

Proof of 2.3. Index the set of unoriented surfaces V (b), by the positive integers in a way that is compatible with increasing bracket length. Then define L_{∞} by setting L(b_{i}) = K_{i} i = 1, …, m and L([b_{1}, b_{2}]) = ∂V (b_{1}) ∩ ∂V (b_{2}). Since V ([b_{1}, b_{2}]) is a Seifert surface for L([b_{1}, b_{2}]), it follows that L_{∞} is an infinite homology boundary link. Thus L is a sublink of a link which is “grown” from L by intersecting Seifert surfaces V_{i} ∩ V_{j} to get new link components and proceeding ad infinitum. In [6, pp. 131–132] it was shown that if this procedure ever terminates or even repeats in the sense that L_{∞} is merely an infinite union of parallels of a finite sublink L′ of L_{∞}, then L′ is an homology boundary link and hence L is a sublink of a finite homology boundary link. The only links known to have vanishing Massey products, namely “fusions of boundary links” [4, 8.10] do indeed admit surface systems with this self-referencing behavior [6, Thm. 2.4].

Since this self-referencing behavior for fusions of boundary links has not been shown in print, let us illustrate with the 2 component ribbon link L = {K_{1}, K_{2}} (a fusion of a 3 component trivial link) in Figure 2.4. The Seifert surfaces V (x), V (y) for L are shown in 2.4. Their intersection K_{3} is shown as a dashed circle in 2.4 and its Seifert surface V ([x, y]) is shown in 2.4. Then V ([x, y]) ∩ V (y) = K_{4} is shown dashed in 2.4. It can be seen that K_{4} and K_{3} cobound an annulus in the complement of interior (V (x) ∪ V (y) ∪ V ([x, y])). Therefore a Seifert surface for K_{4} may be constructed using this annulus and V ([x, y]). Thus it is seen that this may be continued indefinitely in such a way that L_{∞} is merely {K_{1}, K_{2}, K_{3}} together with parallels of K_{3}. For a general fusion of a boundary link an identical situation holds: there exist Seifert surfaces such that L_{∞} is L∪ {parallels of meridians to bands}. If L′ is L together with one “meridian to each band” then it follows almost by definition that L′ is an homology boundary link. For each component K′_{i} of L′_{i} admits a generalized Seifert surface V′_{i} such that V′_{i} ∩ V′_{j} consists only of parallels of {K′_{i} | i ∈ I}. Therefore a larger tubular neighborhood of the K′_{l} will engulf these intersections, leaving generalized Seifert surfaces V′′_{j} which are pairwise disjoint (see Figure 2.5)!

Several remarks are in order about TC as it compares to what one might expect to conclude from the hypothesis of vanishing Massey products. First, condition 4) rules out triple points in the surface systems. Indeed if all Massey products vanish one expects to see triple points occuring in algebraically cancelling pairs, but whether or not it is always Figure 2.4

Figure 2.5

possible to achieve this geometrically is not clear (see Chapter 9 of [4]). Similarly, (3) requires that the intersections of the surfaces are connected. This is a very inconvenient condition in practice. Even for the simplest examples one wants to allow these intersections to be at least a union of parallels of a fixed circle. If one attempts to allow arbitrary disconnectedness, one gets into trouble very quickly as will be seen in Example 2.7.

Theorem 2.6. If L is a finite sublink of an infinite homology boundary link L_{∞}, then the \overline{μ}-invariants of L vanish.

Remarks. The theorem is false under several seemingly reasonable weakenings of the hypothesis. For example it fails if the “components” of an infinite homology boundary link are allowed to be disconnected, that is if we allow K_{n} to be a union of circles. In fact a failure occurs even for such a finite link. Suppose L = {K_{1}, …, K_{5}} where K_{1} to K_{4} are shown in Figure 2.7a. Then each

Figure 2.7

K_{i}, i ≤ 4 bounds a punctured torus V_{i} as in 2.7b such that V_{1} ∩ V_{2} is a circle A, V_{3} ∩ V_{4} is a circle B which link as shown in 2.7c. Let K_{5} = A ∪ B, so K_{5} bounds an annulus disjoint

from ⋃_{i=1}^{4} V_{i}. Then L is an “homology boundary link” except that K_{5} is disconnected. It

is easy to calculate that \overline{μ}(1212) = ±1 for \{K_{1}, K_{2}\} which is a Whitehead Link.

More interestingly, the theorem is false if the surfaces of an infinite homology boundary link are allowed to have an infinite number of boundary components! Consider Figure 2.8 of a cross section of a link with components labelled {K_{1}, K_{2}, … } and surfaces labelled {1, 2, 3, … }. The components converge to a circle labelled ∗. The desired link L_{∞} is

Figure 2.8

obtained by crossing the figure with S^{1}, embedding the resulting link in a solid torus then embedding the solid torus in S^{3} with a full twist. Thus {K_{1}, K_{2}} has linking number one so the conclusion of 2.6 fails. Yet each surface V_{i} is an infinite union of annuli and is a generalized Seifert surface for the K_{i} (except that it has an infinite number of boundary components). This example is especially interesting because meridians can be chosen which go to conjugates of generators under maps to free groups!!

Proof of 2.6. Assume L = {K_{1}, …, K_{m}}. Let G = π_{1}(E(L)) with respect to some basepoint and let u : F ⟨x_{1}, …, x_{m}⟩ → G be defined by u(x_{j}) equals the j^{th} meridian (joined to the basepoint). It suffices to show that there is a homomorphism G → F^{∧}, the nilpotent completion of F, which is an isomorphism on abelianizations. This will be a by-product of our proof of the factorization of Orr’s invariant so we begin an introduction to the latter.

Following Orr, let K_{ω} denote an Eilenberg-Maclane space K(F^{∧}, 1) of F^{∧} ≡ \underleftarrow{lim} F/F_{n} associated to F ⟨x_{1}, …, x_{m}⟩. Let K_{n} denote K(F/F_{n}, 1). Then there are fibrations π′_{n} : K_{n} → K_{n−1} such that \underleftarrow{lim} K_{n} may be identified with K_{ω}. If L is a based link with group G, inducing the meridional map u : F → G, then the \overline{μ}-invariants of L vanish iff u induces an isomorphism on each quotient F/F_{n} → G/G_{n}. Thus u induces a factorization of the

canonical map F ⟶^{i} F^{∧} as follows F ⟶^{u} G ⟶^{π} G^{∧} ⟶^{(û)^{−1}} F^{∧} and consequently induces maps K(F, 1) ⟶^{u_{#}} E(L) ⟶^{ϕ_{#}} K_{ω}, where ϕ = û^{−1} ∘ π. Then ϕ_{#} is extended to S^{3} yielding an element θ_{ω}(L, u) lying in π_{3} of the cofiber of ϕ_{#} ∘ u_{#} = i_{#} [19].

Theorem 2.9. Suppose L is a finite sublink of the infinite homology boundary link L_{∞} and suppose u is any basing which factors through E(L_{∞}). Then there is a compact metric space E, which is an inverse limit of finite 2-complexes, and continuous maps E(L) ⟶^{f̃} E ⟶^{g} K_{ω} inducing a factorization of Orr’s invariant θ_{ω}(L, u) through the

cofiber \overline{E} of the map ⋁_{i=1}^{m} S^{1} ⟶^{f̃∘u} E. \overline{E} is the inverse limit of finite 2-complexes each of

which is homotopy equivalent to a wedge of 2-spheres.

Our strategy to factor Orr’s invariant is to construct the following commutative diagram of spaces.

\begin{matrix}E( & L_{∞}) ⟶^{f} H \\ u ↗ & ↓i  ↙i ↘ g_{H} = λ \\ ⋁_{i=1}^{m} S^{1} ⟶^{u} E & (L) −−→^{f̃} E −−→^{g}  K_{ω} \end{matrix} (2.10)

This will occupy the rest of the chapter. Note that (f̃∘ g)_{∗} : G → F^{∧} as desired to show the \overline{μ}-invariants vanish. Suppose we are given a based link (L, u) such that image (u) ⊂ E(L_{∞}). First we construct the map f, generalizing the well-known result that the exterior of an homology boundary link maps to ⋁_{i=1}^{m} S^{1}. Then we define the space E in which H is

embedded, and the map g. Lastly we define f̃ and compute the Orr invariant.

Recall that to any finite collection of disjointly embedded framed surfaces in S^{3} {Ψ_{i} : V_{i} × [−1, 1] ↪ S^{3}, 1 ≤ i ≤ m} whose boundaries are copies of the longitudes of a link L,

the Pontryagin procedure associates a continuous map p_{m} : E(L) → ⋁_{i=1}^{m} S^{1} by sending the

complement of the images of Ψ_{i} to the wedge point and sending Ψ_{i}(x, t) to (cos πt, sin πt) in the i^{th} circle. Then the inverse image of (1, 0) on the i^{th} circle is V_{i}. Thus if L is an m-component homology boundary link, L admits such a map and the images of some

meridian set normally generates π_{1} ( ⋁_{i=1}^{m} S^{1}). The infinite case yields:

Theorem 2.11. Let L_{∞} be an infinite homology boundary link. Let H be the Hawaiian earrings space, that is \{(x, y) | (x − \frac{1}{n})^{2} + y^{2} = (\frac{1}{n^{2}} ) n ∈ Z_{+}\} in the subspace topology. The infinite surface system {V_{n}} induces a continuous map f : E(L_{∞}) → H such that

f^{−1} ((\frac{2}{n}, 0)) = V_{n} and such that the composition E(L_{∞}) → H → ⋁_{i=1}^{k} S^{1} (the largest k

circles) sends an i^{th} meridian to x_{i} or 0 in H_{1} ( ⋁_{i=1}^{k} S^{1}) according as i ≤ k or i > k.

Moreover, for any n, there is u(n) ∈ Z_{+} such that if i > u(n), the restriction of f to an unbased i^{th} meridian has image disjoint from (\frac{2}{n}, 0).

Proof of 2.11. The following will be used throughout.

Lemma 2.12. If L is an m-component ∞ homology boundary link then for all j ∈ Z_{+} there exist positive integers n(j) and u(j) such that u(j − 1) < n(j) ≤ u(j), n(1) = m,

and ⋃_{i=1}^{n(j)} ∂V_{i} ⊂ ∂E(K_{1} ∪ ⋯ ∪ K_{u(j)}).

Proof of 2.12. Suppose n(j) and u(j − 1) are defined. By 2.4, ⋃_{i=1}^{n(j)} ∂V_{i} is a finite union

of longitudes of components of L so there is some u(j) as claimed. Now set n(j + 1) = u(j) + 1. □

We return to the proof of 2.11. For each j ∈ Z_{+}, 2.12 guarantees a map f_{j} = p_{j} ∘ i

sending E(L_{∞}) ↪^{i} E(K_{1} ∪ ⋯ ∪ K_{u(j)}) ⟶^{p_{j}} ⋁_{i=1}^{n(j)} S^{1} where p_{j} is obtained by the Pontryagin

construction. Let π_{j} or sometimes merely π denote the map ⋁_{i=1}^{n(j)} S^{1} ⟶ ⋁_{i=1}^{n(j−1)} S^{1} obtained

by crushing the last n(j) − n(j − 1) circles. We claim that the f_{n} induce f : E(L_{∞}) → \underrightarrow{lim} ( ⋁_{i=1}^{n} S^{1}). It is only necessary to see that π_{j} ∘ f_{j} = f_{j−1}. If x lies in the complement of

the tubular neighborhoods of {V_{1}, …, V_{n(j)}} then both f_{j} and f_{j−1} send x to the wedge point. If x lies in a tubular neighborhood of V_{i} where i ≤ n(j − 1) then f_{j} and f_{j−1} “agree”. If x ∈ Ψ(V_{i} × [0, 1]) for n(j − 1) < i ≤ n(j) then f_{j−1}(x) is the wedge point. But since f_{j}(x) lies on the i^{th} circle π_{j} ∘ f_{j}(x) is also the wedge point.

It is not difficult to see that H is naturally homeomorphic to \underleftarrow{lim} ⋁_{i=1}^{n} S^{1} via the system

of maps h_{j} : H → ⋁_{i=1}^{n(j)} S^{1} obtained by crushing all but the largest n(j) circles. We shall

identify H with the inverse limit. Thus we have defined a continuous map f : E(L_{∞}) → H as desired. Consider a meridian μ_{i} of the component K_{i} under the map h_{k} ∘ f : E(L_{∞}) →

⋁_{i=1}^{n(k)} S^{1}. Since each V_{j} is a generalized Seifert surface for K_{j}, ∂V_{i} is “algebraically” a single

longitude of K_{i} and if j ≠ i, ∂V_{j} “algebraically” contains no longitudes of K_{i}. Therefore the image of μ_{i} under (h_{k} ∘f)_{∗} on first homology is the generator of the i^{th} circle if i ≤ n(k) and is otherwise zero.

Moreover, if n > 0 there is a positive integer j such that n(j) ≥ n, and so ∂V_{n} ⊂

⋃_{i=1}^{n(j)} ∂V_{i} ⊂ ∂E(K_{1} ∪ ⋯ ∪ K_{u(j)}). Thus if i > u(j) the image of an unbased i^{th} meridian

under f does not go around the n^{th} circle. □

One wonders about the converse to 2.11. Note that the link of 2.7 possesses a map f :

E(L_{∞}) → ⋁_{i=1}^{5} S^{1} while the link of 2.8 does not admit a continuous map f : E(L_{∞}) → H.

We now construct the space E in which H will be naturally embedded. First we establish some notation. Let M_{i} stand for both an unbased i^{th} meridian map M_{i} : S^{1} → E(L_{∞}) (M_{i} = ϕ_{i} |_{∗×∂D^{2}}) and for the same map with range E(K_{1} ∪ ⋯ ∪ K_{s}) when defined. For i ≤ m, we can base M_{i} to agree with the given basing u of L. Let M_{i}(n) denote the composition p_{n} ∘ M_{i} = h_{n} ∘ f ∘ M_{i}. Note that if i > u(j) then M_{i}(n(j)) is the constant map to the wedge point (see 2.11). Let m_{i} (respectively m_{i}(n)) denote an element in the

conjugacy class of M_{i} (respectively M_{i}(n)) in π_{1}(E(L_{∞})) (respectively π_{1} ( ⋁_{i=1}^{n} S^{1})). Note

that π_{j} ∘ M_{i}(n(j)) = M_{i}(n(j − 1)) and π_{∗}(m_{i}(n(j))) = m_{i}(n(j − 1)). Identify π_{1} ( ⋁_{i=1}^{n} S^{1})

with F ⟨x_{1}, …, x_{n}⟩ with the standard generators. Then we remark for later use that for any n and i ≤ n, m_{i}(n) is homologous to x_{i} whereas for i > n, m_{i}(n) is homologous to 0.

Now define the space E_{j} j ≥ 1 by adjoining to ⋁_{i=1}^{n(j)} S^{1} the 2-cells {e_{i}^{2} | m < i ≤ u(j)}

using the maps M_{i}(n(j)) (Recall L = {K_{1}, …, K_{m}}). Define π : E_{j} → E_{j−1} to extend the map π : ⋁_{i=1}^{n(j)} S^{1} → ⋁_{i=1}^{n(j−1)} S^{1} as follows. If i ≤ u(j − 1) then send e_{i}^{2} to e_{i}^{2} by “the identity”

(since M_{i}(n(j − 1)) = π ∘ M_{i}(n(j))). If i > u(j − 1), M_{i}(n(j − 1)) is the constant map to the wedge point so let π(e_{i}^{2}) = ∗ in this case. Setting E = \underleftarrow{lim} E_{j} we see that H is a natural subspace of E.

We shall now inductively construct a family of maps g_{j} : E_{j} → K_{j+1} = K (F/F_{j+1}, 1) such that g_{j−1} ∘ π = π′ ∘ g_{j} where π′ : K_{j+1} → K_{j}, and this will induce g : E → K_{ω} as desired. In addition we shall arrange that g |_{H} ∘f ∘u induces the canonical map F ⟶^{i} F^{∧} on fundamental group.

Lemma 2.14. There are continuous maps λ_{i} : (S^{1}, ∗) → (K_{ω}, ∗) i ∈ Z_{+} such that

1. If i > n(j) then the composition S^{1} ⟶^{λ_{i}} K_{ω} ⟶ K_{j} is the constant map to ∗. Consequently the {λ_{i}} define λ : H ⟶ K_{ω} equal to λ_{i} on the i^{th} circle.

2. If i ≤ m, λ ∘ f ∘ M_{i} represents the image of x_{i} under the map F → F̂.

3. If i > m, λ ∘ f ∘ M_{i} is null-homotopic rel ∗. Therefore so is each ⋁_{i=1}^{n} λ_{i} ∘ M_{i}(n)

(refer to diagram 2.15.)

S^{1} ⟶^{M_{i}} E(L_{∞}) ⟶^{f}H ⟶^{λ} K_{ω} ⟶^{θ_{j}} K_{j}
p_{n} ↘ ↓h_{n}
⋁_{i=1}^{n} S^{1}  ⋁_{i=1}^{n} λ_{i}

M_{i}(n)

Diagram 2.15

Given 2.14 we show how to define {g_{j}}. Let θ_{j} denote the map K_{ω} → K_{j} and λ_{i}^{j} = θ_{j}∘λ_{i}. By induction, we will define g_{j} : E_{j} → K_{j+1} j ≥ 1 such that g_{j−1} ∘ π = π ∘ g_{j} and such

that g_{j} restricted to ⋁_{i=1}^{n(j)} S^{1} is ⋁_{i=1}^{n(j)} λ_{i}^{j+1} which is of course merely θ_{j+1} ∘ λ restricted. Recall

that E_{1} ≡ ⋁_{i=1}^{m} S^{1} ∪ e_{m+1}^{2} ∪ ⋯ ∪ e_{u(1)}^{2} where the two-cells are attached along M_{i}(m) for

m + 1 ≤ i ≤ u(1). By 2.13, m_{i}(m) lies in the commutator subgroup of F ⟨x_{1}, …, x_{m}⟩ so

if we define g_{1} restricted to ⋁_{i=1}^{m} S^{1} ⊂ E_{1} to be θ_{2} ∘ ( ⋁_{i=1}^{m} λ_{i}) = ⋁_{i=1}^{m} λ_{i}^{2} then g_{1} extends to

E_{1} since K_{2} = K (F/F_{2}, 1) has abelian fundamental group. Now suppose g_{j−1} is defined and satisfies the properties stated above. We define g_{j} : E_{j} → K_{j+1} as follows. Set g_{j}

restricted to ⋁_{i=1}^{n(j)} S^{1} to be ⋁_{i=1}^{n(j)} λ_{i}^{j+1}.

Now we check that g_{j−1}∘π = π′∘g_{j} on ⋁_{i=1}^{n(j)} S^{1} where π : E_{j} → E_{j−1} and π′ : K_{j+1} → K_{j}.

We see that π′ ∘ g_{j} is π′ ∘ ⋁_{i=1}^{n(j)} λ_{i}^{j+1} = π′ ∘ θ_{j+1} ∘ ⋁_{i=1}^{n(j)} λ_{i} = θ_{j} ∘ ⋁_{i=1}^{n(j)} λ_{i} = (θ_{j} ∘ ⋁_{i=1}^{n(j−1)} λ_{i}) ∨

( ⋁_{i>n(j−1)} (constant map)) by 2.14.1. But the latter is clearly g_{j−1} ∘ π since π crushes

the circles n(j − 1) < i ≤ n(j). We shall now define g_{j} on a 2-cell e_{i}^{2} of E_{j}. We want π′ ∘ g_{j} to agree with g_{j−1} ∘ π : e_{i}^{2} → K_{j} which is already defined. Since g_{j} | ∂e_{i}^{2} is

θ_{j+1} ∘ ⋁_{i=1}^{n(j)} λ_{i} ∘ M_{i}(n(j)) which is null-homotopic by 2.14.2 (note i > m), so there is some

extension h_{j} : e_{i} → K_{j+1}. Since π_{2}(K_{j}) = 0, π′ ∘ h_{j} is homotopic rel ∂e_{i}^{2} to g_{j−1} ∘ π. Since π′ is a fibration, this homotopy lifts to a homotopy rel ∂e_{i}^{2} from h_{j} to a map called g_{j} : e_{i} → K_{j+1}. This then is the desired g_{j} and covers g_{j−1} ∘ π.

Proof of 2.14. We shall first produce homotopy classes λ_{i} ∈ π_{1}(K_{ω}, ∗) that satisfy 2.14.2 and 2.14.3; then choose particular representatives to achieve 2.14.1.

Let x_{i} : (S^{1}, ∗) → (H, ∗) be the class in π_{1}(H, ∗) represented by the i^{th} circle. Since

π_{1}(H, ∗) modulo the normal subgroup generated by {x_{k} | k > n(j)} is merely π_{1} ( ⋁_{i=1}^{n(j)} S^{1})

detected by the map (h_{n(j)})_{∗}, the image f_{∗}(m_{i}) of a based meridian of the i^{th} component of L_{∞} can be written as a word r_{i}^{j} in {x_{i}, …, x_{n(j)}} modulo this same normal subgroup. Indeed, 2.11 implies that up to conjugation r_{i}^{j} is of the form x_{i}w_{i}^{j}(x_{1}, …, x_{n(j)}) if i ≤ n(j) or w_{i}^{j}(x_{1}, …, x_{n(j)}) if i > n(j), where w_{i}^{j} is a word in the commutator subgroup of F ⟨x_{1}, …, x_{n(j)}⟩. Moreover, by 2.12, for any integer l, if i > u(l) then x_{k} does not occur in w_{i}^{j} for any k ≤ n(l) and any j. We remind the reader that x_{k} would occur precisely when V_{k} had a boundary component on the tubular neighborhood of K_{i}. Finally, since π ∘ m_{i}(n(j)) = m_{i}(n(j − 1)), w_{i}^{j−1} is obtained from w_{i}^{j} by setting x_{i} = e for n(j − 1) < i ≤ n(j).

Let y_{i} 1 ≤ i ≤ m denote the image of the generator x_{i} under the canonical map i : F ⟨x_{1}, …, x_{m}⟩ → F^{∧}. Then we consider the system of equations {r_{i}^{j} = e | i > m, j ≥ 1} ∪ {r_{i}^{j} = y_{i} | i ≤ m, j ≥ 1} over the group F^{∧} in the variables x_{i}. We claim that there exist compatible solutions λ_{i} ∈ F^{∧} in the following sense. If λ_{i}^{j} denotes θ_{j}(λ_{i}), i.e., the image of λ_{i} in F/F_{j+1} = π_{1}(K_{j}) then we shall establish the following by induction:

\underline{P (j)}: There exist elements λ_{i}^{j} ∈ F/F_{j+1} such that

a) λ_{i}^{j} ≡ λ_{i}^{j−1} in F/F_{j},

b) if i > n(j) ≥ m then λ_{i}^{j} = e,

c) if i > m then λ_{i}^{j}w_{i}^{j}(λ_{1}^{j}, …, λ_{n(j)}^{j}) = e in F/F_{j+1},

d) if i ≤ m then r_{i}^{j} = λ_{i}^{j} in F/F_{j+1}.

Once having shown this, by a), the collection {λ_{i}^{j}} defines λ_{i} ∈ F^{∧} which can be realized by maps λ_{i} : S^{1} → K_{ω}. Using b) and the fact that K_{i} → K_{i−1} are fibrations, the λ_{i} may be modified by based homotopy to achieve 2.14.1 “on the nose”. Since K_{ω} = \underleftarrow{lim}(π′ : K_{j+1} → K_{j}), 2.14.1 is precisely what is needed to ensure that the maps λ_{i} induce a continuous map from H ≡ \underleftarrow{lim} ∨S^{1} to K_{ω}. Now, for any i > m and j ≥ 1, the map θ_{j} ∘ λ ∘ f ∘ M_{i} is null-homotopic rel basepoint, because it is represented by a conjugate of the word λ_{i}^{j}w_{i}^{j}(λ_{1}^{j}, …, λ_{n(j)}^{j}) (note λ_{i}^{j} = e if i > n(j)). Similarly if i ≤ m, it is y_{i}^{j}. Thus λ ∘ f ∘ M_{i} is null-homotopic if i > m establishing 2.14.2 and 2.14.3 and completing the construction of g : E ⟶ K_{ω}. It remains only to establish P (j) by induction. Now P (1) holds with λ_{i}^{j} = e for i > m and λ_{i}^{j} = y_{i}^{1} for i ≤ m because n(1) = m and the w_{i}^{j}, being commutators, vanish in the abelian group π_{1}(K_{1}) ≅ F/F_{2}. Now assume P (j − 1). Consider the system of n(j) equations {e = x_{i}w_{i}^{j}(x_{1}, …, x_{n(j)}) | m < i ≤ n(j)} ∪ {y_{i}^{j} = δ_{i}^{j}x_{i}w_{i}^{j}(x_{1}, …, x_{n(j)})(δ_{i}^{j})^{−1} | i ≤ m} in n(j) variables x_{i} over the group F/F_{j+1}. Here δ_{i} is the conjugating element to adjust the right-hand side to be r_{i}^{j}. Such a system has a unique set {λ_{i}^{j} | m < i ≤ n(j)} of solutions over F/F_{j+1} (the proof of Lemma 3 [14] works since the w_{i}^{j} are commutators). Moreover when the equations are projected to F/F_{j}, they are solved by x_{i} = λ_{i}^{j−1} if i ≤ n(j − 1) and x_{i} = e if i > n(j − 1) since w_{i}^{j−1} and δ_{i}^{j−1} are obtained from w_{i}^{j} and δ_{i}^{j} by setting x_{k} = e for k > n(j − 1). Since these solutions are also unique, λ_{i}^{j} ≡ λ_{i}^{j−1} in F/F_{j}. If i > n(j) set λ_{i}^{j} = e. It remains only to show that if i > n(j), λ_{i}^{j}w_{i}^{j}(λ_{i}^{1}, …, λ_{n(j)}^{j}) = e in F/F_{j+1}. since i > n(j) > u(j − 1) (by 2.12), w_{i}^{j} contains no occurences of x_{k} for k ≤ n(j − 1). Therefore w_{i}^{j}(λ_{1}^{j}, …, λ_{n(j)}^{j}) is a product of conjugates of commutators [λ_{α}^{j}, λ_{β}^{j}] where n(j − 1) < α, β ≤ n(j). Since λ_{α}^{j−1} = e = λ_{β}^{j−1} in F/F_{j} for such α, β, λ_{α}^{j−1} and λ_{β}^{j−1} lie in F_{j}/F_{j+1} and so their commutator vanishes in F/F_{j+1}. This concludes the proof of P (j) and hence 2.14 and the definition of g : E ⟶ K_{ω}. □

Now, referring to 2.10, we construct the map f̃: E(L) ⟶ E such that i ∘ f = f̃ ∘ i. Let f̃ = \underleftarrow{lim} f̃_{j} where f̃_{j} : E(L) ⟶ E_{j} is as follows. On E(L_{∞}) let f̃_{j} agree with f (for this proof let f stand for the composite E(L_{∞}) ⟶^{f} H ⟶^{i} E ⟶ E_{j}). On the tubular neighborhood ϕ_{i}(S^{1} × D^{2}) of K_{i} (i > m), let f̃_{j} be the constant map to ∗ if i > u(j); and if i ≤ u(j) let it send ϕ_{i}(S^{1} ×D^{2}) to ϕ_{i}({1}×D^{2}) and identify this with the 2-cell e_{i}^{2} of E_{j}. One checks that f̃_{j−1} = π′_{j} ∘f̃_{j}. To show f̃_{j} is continuous it suffices to show it is continuous

on the closed sets A = E_{L} − ⋃_{i=1}^{u(j)} int (ϕ_{i}(S^{1} × D^{2})) and B = ⋃_{i=1}^{u(j)} ϕ_{i}(S^{1} × D^{2}). The latter

restriction is clearly continuous. Suppose x_{0} ∈ A. The restriction to A is clearly continuous at x_{0} if x_{0} ∈ E_{L_{∞}} − ⋃_{u(j)}^{∞} ϕ_{i}(S^{1} × D^{2}) since the latter is open in E(L_{∞}) and f̃_{j} agrees with

f there. It is also clearly continuous at x_{0} if x_{0} lies in some int (ϕ_{i}(S^{1} × D^{2})). Suppose x_{0} ∈ ϕ_{i}(S^{1} × S^{1}) and x_{n} → x_{0} where x_{n} ∈ A. If x_{n_{j}} is the subsequence of elements of E(L_{∞}) then x_{n_{j}} → x_{0} in E(L_{∞}) so f̃_{j}(x_{n_{j}}) = f(x_{n_{j}}) ⟶ f(x_{0}) = f̃_{j}(x_{0}). Thus given any neighborhood U of f_{j}(x_{0}) in E_{j}, there is N s.t. if n > N and x_{n} ∈ E(L_{∞}) then

f̃_{j}(x_{n}) ∈ U. Since A = E(L_{∞}) ⋃_{i>u(j)} ϕ_{i}(S^{1}×D^{2}), if x_{n} ∈ A−E(L_{∞}) then x_{n} ∈ ϕ_{i}(S^{1}×D^{2})

where i > u(j) so f̃_{j}(x_{n}) = ∗ = f̃_{j}(x_{0}) by definition. Thus f̃_{j}(x_{n}) ⟶ f̃_{j}(x_{0}). Since A is metrizable, f̃_{j} is continuous on A. Hence f̃_{j} itself is continuous. This completes the construction of f̃: E(L) → E as in 2.10.

We return to the factorization of Orr’s ω-invariant. We must suppose that u : ⋁_{i=1}^{m} S^{1} ⟶

E(L) is a basing which avoids L_{∞}. Then we have constructed the entire diagram 2.10. Moreover if G = π_{1}(E(L)) and Ψ = (g ∘ f̃)_{∗} : G ⟶ F^{∧} then, since we have arranged that Ψ ∘ u : F ⟶ F^{∧} is i : F ⟶ F^{∧}, it follows that Ψ^{∧} ∘ μ^{∧} = i^{∧} = identity. Consequently Ψ^{∧} = (μ^{∧})^{−1} and the Orr invariant factors through the cofiber of the map f̃∘ u : K(F, 1) ⟶ E (since Ψ_{#}^{∧} = g ∘ f̃). Let \overline{E} stand for this cofiber. More specifically

we may take \overline{E} to be E ⋃_{i=1}^{m} e_{i}^{2} where the 2-cell e_{i}^{2} is attached via i ∘ f ∘ M_{i} (the image of

the meridian of the i^{th} component). Then we see that \overline{E} ≡ \underleftarrow{lim} \overline{E}_{j} where \overline{E}_{j} = E_{j} ⋃_{i=1}^{m} e_{i}^{2}

where e_{i}^{2} is attached along π ∘ i ∘ f ∘ M_{i} which is merely M_{i}(n(j)). Thus \overline{E} is the inverse

limit of finite 2-complexes which we claim are simply-connected. Since \overline{E}_{j} = ⋁_{i=1}^{n(j)} S^{1} ∪

{e_{i}^{2} | 1 ≤ i ≤ u(j)} using the maps M_{i}(n(j)), it suffices to show that π_{1} ( ⋁_{i=1}^{n(j)} S^{1})

is normally generated by {M_{i}(n(j)) | 1 ≤ i ≤ u(j)}. Recalling the Pontryagin map

p : E(K_{1} ∪ ⋯ ∪ K_{u(j)}) ⟶ ⋁_{i=1}^{n(j)} S^{1}, since π_{1} (E(K_{1} ∪ ⋯ ∪ K_{u(j)})) is normally generated

by the meridians M_{i}, 1 ≤ i ≤ u(j), it suffices to show that p_{∗} is surjective on π_{1}. We now

argue that ⋃_{i=1}^{n(j)} V_{i} does not separate E(K_{1} ∪ ⋯ ∪ K_{u(j)}). Let V̂_{i} be the space obtained

by identifying each component of ∂V_{i} which is a longitude of K_{s} to a single circle, for

each 1 ≤ s ≤ u(j). Let V̂ be ⋃_{i=1}^{n(j)} V_{i} with similar identifications. Then V̂ is a natural

subspace of S^{3} and E(K_{1} ∪ ⋯ ∪ K_{u(j)}) − ⋃_{i=1}^{n(j)} V_{i} is homotopy-equivalent to S^{3} − V̂. By

Alexander Duality, \overline{H}^{0}(S^{3} −V̂) ≅ H_{2}(V̂). Since the V_{i} are (connected!) generalized Seifert surfaces, V̂_{i} is obtained from a surface with a single boundary component by identifying sets of homologically linearly independent circles. Thus it follows that H_{2}(V̂_{i}) is zero and that H_{2}(V̂) ≅ 0 follows from an extension of this reasoning, since V̂ is a quotient of

∐_{i=1}^{n(j)} V̂_{i}. Therefore, ⋃_{i=1}^{n(j)} V_{i} does not separate E(K_{1} ∪ ⋯ ∪ K_{u(j)}). For each 1 ≤ i ≤ n(j),

choose an interval fiber of the tubular neighborhood of V_{i} and choose a path disjoint from ⋃_{i=1}^{n(j)} V_{i} connecting its endpoints. This loop then maps under p_{∗} to a generator of π_{1} of

the i^{th} circle. Hence \overline{E} is an inverse limit of finite 2-complexes each of which is homotopy equivalent to a wedge of 2-spheres. This completes the proofs of 2.6 and 2.9.

Chapter 3: Remarks on the spaces E and \overline{E}.

We have seen that \overline{E} is the inverse limit of finite 2-complexes, each of which is homotopyequivalent to a wedge of 2-spheres. Certainly not every such inverse limit space is con-

tractible. For example suppose we define X_{j} = ⋁_{i=1}^{j} S^{1} ∪ e_{1}^{2} ∪ e_{2}^{2} ∪ ⋯ ∪ e_{j}^{2} where e_{i}^{2} i ≠ j

attaches to ⋁_{i=1}^{j} S^{1} along a path traversing the i^{th} circle counter-clockwise and e_{j}^{2} attaches

along the j^{th} circle clockwise. Then let \overline{X}_{j} = X_{j} ∪ e_{0}^{2} where ∂e_{0}^{2} is the 1st-circle clockwise, and define \overline{X}_{j} ⟶^{π} \overline{X}_{j−1} to send e_{j}^{2} to the wedge point, e_{i}^{2} 0 ≤ i < j − 1 to itself, and e_{j−1}^{2} in \overline{X}_{j} to its companion e_{j−1}^{2} in \overline{X}_{j−1}. Then clearly \underleftarrow{lim} \overline{X}_{j} ≃ \overline{X}_{j} ≃ S^{2} and π_{∗} is an isomorphism on π_{2}. Thus \underleftarrow{lim} H_{2}(\overline{X}_{j}) = H_{2}(\underleftarrow{lim} \overline{X}_{j}) ≅ Z. Therefore the way the 2-cells are attached to \overline{E}_{j} is very important. In particular in our situation we at least have the following:

Theorem 3.1. For any j ∈ Z^{+}, there exists a positive integer d = d(j) such that the map \overline{E}_{d} → \overline{E}_{j−1} is null-homotopic. In particular \underleftarrow{lim} H_{∗}(\overline{E}_{j}) = \underleftarrow{lim} π_{∗}(\overline{E}_{j}) = 0 if ∗ ≥ 1.

Proof. Since each \overline{E}_{j} is a wedge of 2-spheres it suffices to show H_{2}(\overline{E}_{d}) → H_{2}(\overline{E}_{j−1}) is

zero. Recall that \overline{E}_{d} = ⋁_{i=1}^{n(d)} S^{1}∪\{e_{i}^{2} | 1 ≤ i ≤ u(d)\} where n(d), u(d) are as in 2.12 and e_{i}^{2} is

attached along M_{i}(n(d)) as defined above 2.14. By those remarks M_{i}(n(d)) is homologous

to x_{i} if 1 ≤ i ≤ n(d) and 0 if i > n(d). Thus the image of H_{2}(\overline{E}_{d}) → H_{2} (\overline{E}_{d}, ⋁_{i=1}^{n(d)} S^{1})

is generated by {e_{i}^{2} | n(d) < i ≤ u(d)}. By 2.12, given j, there is an integer d such that n(d) > u(j − 1). Then under the map π : \overline{E}_{d} → \overline{E}_{j} a cell e_{i}^{2} above maps either to the wedge point or to the cell e_{i}^{2} attached along M_{i}(n(j)) in \overline{E}_{j}. But since i > n(d) > u(j − 1) the cell e_{i}^{2} of \overline{E}_{j} maps to the wedge point in \overline{E}_{j−1} by the definition above 2.14. Therefore the image H_{2}(\overline{E}_{d}) → H_{2}(\overline{E}_{j−1}) is zero. □

This may be compared with the result of Bousfield and Kan that \underleftarrow{lim} H_{q}(F/F_{n}) = 0 for q ≥ 2 which implies the results of 3.1 for the tower → \overline{K}_{j} → \overline{K}_{j−1} (where \underleftarrow{lim} \overline{K}_{j} = \overline{K}_{ω}) (see Theorem 1.2 of [5]). Thus 3.1 merely reflects our claim that the tower {E_{j}} is basically a 2 skeleton of the tower {K(F/F_{n}, 1)}. If this is precisely true then Bousfield’s result that H_{2}(F^{∧}; Q) is uncountable is discouraging since then, by cardinality considerations, most of these elements could not contribute to a non-zero Orr invariant [2; Prop. 4.4]. Still, the concreteness of the spaces \overline{E} leads us to hope that they can be understood and perhaps even facilitate an understanding of H_{∗}(F^{∧}). Understanding even a single non-zero element of π_{3}(\overline{E}) would very likely lead to its realization and a new type of linking phenomenon.

We close with some interesting remarks and questions.

We do not know if \overline{E} is simply-connected (as K_{w} is) but expect this to be the case. For the surface systems that arise in 2.3 and especially TC part 3, one sees that the most fundamental example of a space E arises from imagining an m-component link with Seifert surfaces each pair of which intersects in a circle and that each of these circles possesses a (true) Seifert surface which intersects each previous one in a single circle, etcetera. The surfaces of this imagined system and hence the circles of the Hawaiian Earrings are indexed by the simple commutators of F ⟨x_{1}, …, x_{m}⟩ which are ordered compatibly with commutator length. Each component c(b_{1}, b_{2}) = ∂(V (b_{1}, b_{2})) of this hypothetical L_{∞} arises precisely as V (b_{1}) ∩ V (b_{2}) (= −(V (b_{2}) ∩ V (b_{1})) and so the corresponding meridian f_{∗}(M_{(b_{1},b_{2})}) ∈ π_{1}(H) is merely x_{(b_{1},b_{2})}[x_{(b_{1})}, x_{(b_{2})}]. In this standard situation the map λ : H → K_{ω} is defined by sending λ_{i} to the i^{th} simple commutator in the ordering (compare 2.14). It is then easy to show that λ induces an epimorphism λ_{∗} : π_{1}(H) ↠ F^{∧}. The group π_{1}(H) is itself fascinating. It is ω-milpotent like F^{∧} but is locally free, unlike F^{∧}. Moreover if {g_{i}} is any sequence of elements of π_{1}(H) which converges to 1 in the obvious inverse-limit topology, then there is an endomorphism of π_{1}(H) sending x_{i} → g_{i} [11]. It would be nice to factor Levine’s refinement \overline{θ}_{ω} of Orr’s invariant, but one can see from the proof of 2.14 that an infinite link leads to an infinite number of equations whereas Levine’s algebraic closure \overline{F} ⊂ F^{∧} is defined using sets of finite number of equations. Still the similarities are striking and bear further study.

Acknowledgments.

Partially supported by N.S.F. grants DMS-8903514, DMS-9100254.

First announced in special session of American Mathematical Society, Chicago, Illinois, May 1989

References

1. M. Barrat and J. Milnor, An example of anomalous singular homology, Proc. Amer. Math. Soc. (1962), 293–297.

2. A.K. Bousfield, Homological Localization Towers for Groups and π-Modules, Memoirs A.M.A. 10 (1977), 186.

3. S. Cappell and J. Shaneson, Link cobordism, Comment. Math. Helv. 55 (1980), 20–49.

4. T.D. Cochran, Derivatives of Links: Milnor’s concordance invariants and Massey’s products, Memoirs A.M.S. #427, Volume 84, (1990), American Math. Soc., Providence, Rhode Island.

5. , Link concordance invariants and homotopy theory, Inventiones Math. 90 (1987), 635–645.

6. , Localization and finiteness in link concordance, Topology and its Applications 32 (1989), 121–133.

7. T.D. Cochran and J. Levine, Homology boundary links and the Andrews-Curtis conjecture, Topolgoy 30 (1991), no. 2, 231–239.

8. T.D. Cochran and K. Orr, Not all links are concordant to boundary links, Bull. Amer. Math. Soc. 23 (1990), no. 1, 99–106.

9. , Not all links are concordant to boundary links, preprint 1991, Rice University.

10. H.B. Griffiths, Infinite products of semi-groups and local connectivity, Proc. London Math. Soc. 6 (1956), 455–485.

11. Higman, Unrestricted free products and varieties of topological groups, Journal London Math. Soc. (1952), 73–81.

12. J.A. Hillman, Alexander Ideals of Links, Springer-Verlag, Berlin/New York, 1981.

13. D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431–449.

14. J.P. Levine, Link concordance and algebraic closure of groups, Commentarii Math. Helv. 64 (1989), 236–255.

15. , Link concordance and algebraic closure II, Inventiones Math. 96 (1989), 571–592.

16. J. Milnor, Isotopy of Links, in: Algebraic Geometry and Topology: A Symposium in Honor of Solomon Lefshetz, Princeton Univ. Press, Princeton, N.J., 1957, pp. 280–306.

17. J.W. Morgan and I. Morrison, A Van Kampen Theorem for Weak Joins, Proc. London Math. Soc. 53 (1986), 562–576.

18. K. Orr, New link invariants and applications, Comment. Math. 62 (1987), 542–560.

19. , Homotopy invariants of link concordance, Inventiones Math. 95 (1988), 379–394.

20. R. Porter, Milnor’s \overline{μ}-invariants and Massey products, Trans. Amer. Math. Soc. 257 (1980), 39–55.

21. N. Smythe, Boundary links, in: R.H. Bing, ed., Topology Seminar: Wisconsin 1965, Ann. of Math. Study 60 (Princeton Univ. Press, Princeton, N.J., 1966) 69–72.

22. V. Turaev, The Milnor invariants and Massey products, in: Studies in Topology II, Zap. Nanc. Sem. Lengrad Ot. Mat. Kogo, oi Stet. Acad. Nauk. USSR 66 (1976). (Translation in: J. Soviet Math.).