Reflections on C. T. C. Wall’s work on cobordism

Jonathan Rosenberg^{∗}

December 11, 1997

1 Introduction

This book is intended to fulfill two main functions—to celebrate C. T. C. Wall’s contributions to topology, and to discuss the state of surgery theory today. This article concentrates on Wall’s first contribution to topology, still of great significance: the completion of the calculation of the cobordism ring Ω defined by René Thom. This subject, while not directly a part of surgery theory, is still vital to it. Surgery theory classifies manifolds (smooth, PL, or topological) within a given homotopy type, starting from the observation that one can construct new manifolds from old ones by means of surgery. It is obvious that if two manifolds are related by surgery, then they are cobordant, and the fundamental theorem of Morse theory, brought to the fore in this context by Milnor, shows that two (smooth) manifolds are cobordant if and only if they are related by a sequence of surgeries. It turns out that, in some sense, refinements of this idea are basic to surgery theory. The notions of normal cobordism of maps, algebraic cobordism of algebraic Poincaré complexes, and so on, play a major role. Also, from a historical point of view, Wall’s interest in the surgery classification of manifolds grew out of his earlier work on cobordism. But there are also more subtle reasons for studying oriented bordism in connection with surgery theory. For example, the symmetric signature of Mishchenko-Ranicki may be viewed as a natural transformation from oriented bordism to symmetric L-theory ([Ran1], §7.1; [Ran2]), and Wall’s work on the structure of the oriented bordism spectrum may be used to deduce facts about the structure of surgery spectra, such as the fact that they are Eilenberg-MacLane when localized at 2 [TayW].

To explain Wall’s contribution we first need to say a few words about Thom’s work.

∗Partially supported by NSF Grant # DMS-96-25336.

2 Thom’s work on cobordism

Two closed (smooth) n-manifolds M_{1}^{n} and M_{2}^{n} are called cobordant if there is a compact manifold with boundary, say W^{n+1}, whose boundary is (diffeomorphic to) the disjoint union of M_{1} and M_{2}. It is trivial to note that this is an equivalence relation, and that with the addition operation defined by the disjoint union of manifolds, the set 𝔑_{n} of cobordism classes of closed n-manifolds is an abelian group. This group has exponent 2, since the boundary of M^{n} × [0, 1] is (diffeomorphic to) M ⨿ M. Furthermore, 𝔑 = ⨁_{n=0}^{∞} 𝔑_{n} is a commutative graded algebra over ℤ/2, with multiplication defined by the Cartesian product of manifolds. Thom [Thom] had succeeded in computing this ring by reducing the calculation to a problem in homotopy theory. This is done using the famous Thom-Pontrjagin construction. For N sufficiently large, a closed n-manifold M^{n} may (by the easy part of the Whitney Embedding Theorem) be embedded smoothly in ℝ^{n+N}, and (provided N is large enough) any two such embeddings are isotopic to one another. Given such an embedding, choose a closed tubular neighborhood T of M in ℝ^{n+N}; this is diffeomorphic to the unit disk bundle D(E) of the normal bundle E for the embedding. By the classification theorem for vector bundles, E is obtained by pulling back the universal rank-N vector bundle E_{N}^{u} = EO(N) ×_{O(N)} ℝ^{N} over BO(N), via a classifying map f : M → BO(N) whose homotopy class is independent of all choices made. Let MO_{N} be the space obtained from the unit disk bundle D(E_{N}^{u}) of E_{N}^{u} by collapsing the sphere bundle S(E_{N}^{u}) to a point; this is called the Thom space of the vector bundle E_{N}^{u}. Map S^{n+N} = ℝ^{n+N} ∪ {∞} to T/∂T by collapsing the complement of the interior of T to a point, and then map to MO_{N} by the obvious map obtained from the lift of f to the bundle map f̃: E → E_{N}^{u}. The result is a map t_{M} : S^{n+N} → MO_{N} whose homotopy class only depends on the cobordism class of M as a smooth manifold, since a cobordism W between M and M′ gives rise (by a similar construction with W in place of M) to a homotopy between the maps t_{M} and t_{M′}. Thom showed in his fundamental paper [Thom] that in this way one obtains a natural isomorphism between 𝔑_{n} and π_{n}(MO) = lim_{N→∞} π_{n+N}(MO_{N}). The inverse of the map is obtained by transversality: given t : S^{n+N} → MO_{N}, we make t transverse to the 0-section 0 of E_{N}^{u} (this requires a little technical fiddling since MO_{N} isn’t quite a manifold), and we recover M^{n} up to cobordism as t^{−1}(0). (For more details, an excellent reference is [Stong].)

It’s convenient to rework Thom’s work in more modern language. There are maps ΣMO_{N} → MO_{N+1} coming from the fact that the product with ℝ of the universal bundle E_{N}^{u} over BO(N) is pulled back from the universal bundle E_{N+1}^{u} over BO(N + 1). Thus the sequence of spaces {MO_{N}} forms a spectrum MO in the sense of stable homotopy theory (the best reference on these for the beginner is [Ad2], Part III), and Thom’s result shows that π_{∗}(MO) ≅ 𝔑_{∗}. In fact, MO is even a ring spectrum, with product MO ∧ MO → MO coming from the Whitney sum of vector bundles, and π_{∗}(MO) ≅ 𝔑_{∗} is a ring isomorphism. Finally, to restate Thom’s work in the modern language of spectra, Thom was able to show that MO is a generalized Eilenberg-MacLane spectrum, and that 𝔑 is a polynomial ring over ℤ/2, with one generator x_{k} in each degree k ≥ 1 not of the form 2^{j} − 1. (Thus the first few generators are in dimensions 2 and 4, corresponding to the manifolds ℝℙ^{2} and ℝℙ^{4}.) The fact that the spectrum MO is of Eilenberg-MacLane type means that the mod-2 Hurewicz map π_{∗}(MO) → H_{∗}(MO; ℤ/2) is an injection. This has the geometric interpretation that all information about (unoriented) cobordism comes from mod-2 homology, and thus that a manifold is determined up to cobordism by its Stiefel-Whitney characteristic numbers. Since these only depend on the homotopy type of the manifold, homotopy equivalent manifolds are necessarily cobordant.

Now Thom had observed in [Thom] that one can do something quite similar with oriented manifolds. Two oriented n-manifolds M_{1}^{n} and M_{2}^{n} are called oriented cobordant if there is a compact oriented manifold with boundary, say W^{n+1}, whose boundary is (diffeomorphic to) the disjoint union of M_{1} and M_{2}, and if the orientation on W induces the given orientation on M_{1} and the reverse of the given orientation on M_{2}. This is again an equivalence relation, since the standard orientation on M × [0, 1] induces the given orientation on M × {0} and the reverse of the given orientation on M ×{1}. So oriented cobordism classes of n-manifolds form an abelian group Ω_{n} under disjoint union, with the inverse of the class of M given by the class of \overline{M}, the same manifold as M, but with opposite orientation. One finds that Ω = ⨁_{n=0}^{∞} Ω_{n} is a graded commutative graded ring, the oriented cobordism ring, with multiplication defined by the Cartesian product of (oriented) manifolds. Thom’s work (slightly rephrased) showed that the Thom-Pontrjagin construction works as before, to give an isomorphism Ω_{∗} ≅ π_{∗}(MSO), where the ring spectrum MSO is defined the same way as MO, but using the universal oriented bundle E_{N}^{u,o} over BSO(N) in place of the universal unoriented bundle E_{N}^{u} over BO(N).

Rationally, Thom had no trouble in computing Ω, for Ω_{∗} ⊗_{ℤ} ℚ ≅ π_{∗}(MSO)⊗_{ℤ} ℚ, and rational stable homotopy is the same thing as rational homology. Furthermore, by the Thom isomorphism for oriented vector bundles (also developed in [Thom]), H̃_{k+N}(MSO_{N}; ℚ) ≅ H_{k}(BSO(N); ℚ). Since H^{∗}(BSO; ℚ) is the power series algebra in the rational Pontrjagin classes, it easily follows that rationally, oriented cobordism is determined by the Pontrjagin characteristic numbers. Thom deduced from this that Ω ⊗_{ℤ} ℚ is a polynomial algebra on the classes of the complex projective spaces ℂℙ^{2j}, j ≥ 1. The problem left open by Thom and solved by Wall was the integral calculation of Ω.

3 Wall’s work on cobordism

The first major step in the integral calculation of Ω was taken by Milnor, who proved:

Theorem 3.1 [Mil] Ω has no torsion of odd order.

But the hardest part of the calculation, involving the 2-primary torsion, still remained. Some work on this had been done by Rokhlin, but not all of it was correct. Wall’s original method for completing the calculation of Ω was both ingenious and geometric. He introduced a new cobordism theory 𝔚 = ⨁_{n=0}^{∞} 𝔚_{n}, where 𝔚_{n} consists of cobordism classes of pairs (M^{n}, f), where M^{n} is a closed (unoriented) n-manifold, f : M → S^{1}, and the pull-back under f of the canonical generator of H^{1}(S^{1}; ℤ) reduces mod 2 to w_{1}(M). Two such pairs (M_{1}^{n}, f_{1}) and (M_{2}^{n}, f_{2}) are equivalent if there is a compact manifold with boundary W^{n+1} and a map f : W → S^{1} such that ∂W = M_{1} ⨿ M_{2}, f restricts on M_{j} to f_{j}, and the pull-back under f of the canonical generator of H^{1}(S^{1}; ℤ) reduces mod 2 to w_{1}(W). (Example: suppose M = S^{1} and f : M → S^{1} is a map of degree 2. Then the pair (M, f) defines an element of 𝔚, since the degree of f vanishes mod 2. This element of 𝔚 is trivial, since M bounds a Möbius strip W, and f extends to a map f̃ : W → S^{1}, inducing an isomorphism on π_{1} and pulling back the generator on H^{1}(S^{1}; ℤ/2) to w_{1}(W).) Note that if M^{n} is any closed n-manifold, one can find an f : M → S^{1} for which (M^{n}, f) represents a class in 𝔚_{n} if and only if w_{1}(M) is the reduction of an integral class, or if and only if βw_{1} = 0, where β : H^{1}(M; ℤ/2) → H^{2}(M; ℤ) is the Bockstein operator. Since the reduction mod 2 of the Bockstein coincides with the Steenrod operation Sq^{1}, this condition implies Sq^{1}w_{1} = 0, or w_{1}^{2} = 0. If we drop the w_{1} condition, there is a forgetful map 𝔚_{n} → 𝔑_{n}(S^{1}) to the cobordism classes of pairs (M^{n}, f), where M^{n} is a closed (unoriented) n-manifold, and f : M → S^{1}. (The equivalence relation on these pairs is as before, but without the w_{1} condition.) One can easily check (cf. [Atiyah]; this is also related to results in [Wall4]) that X ↦ 𝔑_{n}(X) defines a homology theory, which can be computed as 𝔑_{n}(X) ≅ π_{∗}(X_{+} ∧ MO). (The subscript _{+} denotes the addition of a disjoint basepoint, which comes from the fact that we are using an unreduced homology theory here.) Thus 𝔑_{n}(S^{1}) ≅ 𝔑_{n} ⊕ 𝔑_{n−1}, and the class of a pair (M^{n}, f) in 𝔑_{n}(S^{1}) is determined by characteristic numbers of M involving the Stiefel-Whitney classes and the pull-back f^{∗}(x) under f of the generator x of H^{1}(S^{1}; ℤ/2). When the pair (M^{n}, f) comes from 𝔚_{n}, then f^{∗}(x) = w_{1}, so all these characteristic numbers are thus ordinary Stiefel-Whitney numbers. It is not hard to see from this that the forgetful map 𝔚_{n} → 𝔑_{n} is injective, and identifies 𝔚_{n} with the classes in 𝔑_{n} of manifolds for which all Stiefel-Whitney numbers involving w_{1}^{2} vanish.

Now we can explain Wall’s approach [Wall1, Wall2] to the calculation of the cobordism ring Ω. (However we will make use of some simplifications subsequently found in the proofs. Still another variant on the argument can be found in [Ad1].) The idea is to set up an exact triangle

[figure: Ω; 2; Ω; s; 𝔚,; ∂]  (1)

where 2 denotes multiplication by 2 on the abelian group Ω, s is the forgetful map from Ω to 𝔚 (an oriented manifold has w_{1} = 0, hence defines a class in 𝔚), and ∂ is the boundary map (decreasing degree by 1) obtained by “dualizing w_{1}.” (In other words, given M^{n} with w_{1}(M) the reduction of an integral cohomology class, let N^{n−1} ⊂ M^{n} represent the mod-2 homology class which is the Poincaré dual (mod 2) to w_{1}(M). This manifold inherits an orientation from the implicit map f : M → S^{1}. Furthermore, N as an oriented manifold is well-defined up to cobordism.) Wall’s original proof of the exactness of the triangle was somewhat complicated, but he gave a simpler and more geometric proof in [Wall5]. The first part is the exactness of Ω →^{s} 𝔚 →^{∂} Ω. We have ∂ ∘ s = 0, since if M is an oriented manifold, then dualizing w_{1} gives the empty submanifold. In the other direction, if M^{n} gives a class in 𝔚 and ∂[M^{n}] = 0, that means that the Poincaré dual of w_{1}(M) is null-cobordant, so that all the Stiefel-Whitney numbers of M involving w_{1} vanish. From Thom’s description of generators of 𝔑, this implies that M is cobordant to something orientable. A slightly harder part of the proof is the exactness of Ω →^{2} Ω →^{s} 𝔚. That the composite of the two arrows is trivial is clear, since 𝔚 ⊂ 𝔑, which as an abelian group has exponent 2. We need to show that if [M] ∈ Ω_{n} maps to 0 in 𝔚_{n}, then [M] is divisible by 2. The condition that [M] ∈ Ω_{n} maps to 0 in 𝔚_{n} means that M is the boundary of an unoriented manifold W^{n+1}, and that there is a map f : W → S^{1} whose restriction to M is null-homotopic and such that f^{∗}(x) = w_{1}(W). We may assume that f sends M to a point in S^{1}, say 1. Then the inverse image under f of a regular value ≠ 1 of f is a compact submanifold A^{n} which doesn’t meet the boundary of W and thus is closed. Furthermore, A is “dual” to w_{1}(W) and has trivial normal bundle. Removing an open tubular neighborhood of A from W, we obtain a compact oriented manifold V with boundary ∂V ≅ A ⨿ A ⨿ M, so [M] is divisible by 2 in Ω_{n}. Finally, we check exactness of 𝔚 →^{∂} Ω →^{2} Ω. That 2 ∘ ∂ = 0 is obvious since 𝔚 is a group of exponent 2. On the other hand, if M^{n} is closed and oriented and 2[M] = 0 in Ω_{n}, choose W^{n+1} oriented with boundary ∂W ≅ M ⨿ M, then glue the two copies of M together to get a closed (non-oriented) manifold V^{n+1} in which M is dual to w_{1}. One can easily show that in this case w_{1}(V) comes from a map to S^{1}, so that V represents an element of 𝔚_{n+1} with ∂[V] = [M].

From the exact triangle (1), Wall deduced his main result on the cobordism ring:

Theorem 3.2 (Wall [Wall2]) All torsion in Ω is of order 2. The oriented cobordism class of an oriented closed manifold is determined by its Stiefel-Whitney and Pontrjagin numbers.

This is not just a simple corollary of the exact triangle (1), and Wall needed some additional ingredients. One of these was a Milnor’s result quoted above. Another was a more detailed analysis of 𝔑 and 𝔚, based in part on [Dold]. Wall showed that 𝔚 is a subalgebra of 𝔑 (this is obvious from the description above involving vanishing of Stiefel-Whitney numbers involving w_{1}^{2}), and that one can choose the polynomial generators x_{k} (k ≠ 2^{j} −1) for 𝔑 so that x_{k} is represented by an oriented manifold for k odd and 𝔚 is the subalgebra generated by x_{k}, k not of the forms 2^{j} or 2^{j} − 1, and by the x_{2^{j}}^{2}. (This is done using variants on manifolds constructed by Dold [Dold]. The class x_{2^{j}}^{2} is represented by a complex projective space.) Finally, Wall showed that with these choices of generators, the map s ∘ ∂ : 𝔚 → 𝔚 is the derivation sending x_{2j} ↦ x_{2j−1} (j not a power of 2) and killing the classes x_{2j−1} and x_{2^{j}}^{2}. The main step in the proof of Theorem 3.2 is therefore:

Lemma 3.3 (Wall [Wall2]) Ω has no torsion of order 2^{t}, t > 1.

Proof. Suppose c ∈ Ω_{n} and 2^{t}c = 0, 2^{t−1}c ≠ 0, t > 1. Then 2·2^{t−1}c = 0, so by (1), 2^{t−1}c = ∂y, for some y ∈ 𝔚_{n+1}. Also, since t > 1, 2^{t−1}c = 2 · 2^{t−2}c and hence, again by (1), s (2^{t−1}c) = 0. So s∘∂y = 0. From the description of s ∘ ∂ above, this means y is a sum of monomials in the x_{k}’s in which no x_{2j} (j not a power of 2) occurs to odd order. But each x_{k}^{2} is represented by an oriented manifold (that’s because one has alternative generators for 𝔑 in even degrees given by real projective spaces, whose squares are cobordant to complex projective spaces ([Wall2], Proposition 3)), hence y ∈ im Ω and thus ∂y = 0 by (1), i.e., 2^{t−1}c = 0. This is a contradiction. □

Note as well that once one has proved this lemma, it follows that the forgetful map Ω → 𝔑 is injective on the torsion in Ω, and thus that the torsion classes in Ω are detected by Stiefel-Whitney numbers. Indeed, since all torsion in Ω is of order 2, the torsion injects into Ω/2Ω, which then injects into 𝔚 ⊂ 𝔑 by (1). Wall went further than this and computed the multiplicative structure of Ω; this depends on the fact that in the triangle (1), 𝔚 is an algebra, s is a ring homomorphism, and s ∘ ∂ is a derivation. (One may find details in Chapter IX of [Stong].)

Aside from the above proof, there is another way of approaching Theorem 3.2, which to the modern algebraic topologist might seem more natural. Namely, one can work entirely homotopy-theoretically, and bypass direct geometrical considerations altogether. The problem is one of computing π_{∗}(MSO), so one can localize at a prime p and apply the Adams spectral sequence ([Ad2], Part III, and [Swit]) to compute π_{∗}(MSO) from H_{∗}(MSO; ℤ/p), viewed as a comodule over the dual mod-p Steenrod algebra A_{∗}^{p}. The Hurewicz map π_{∗}(MSO) → H_{∗}(MSO; ℤ/p) appears as an edge homomorphism of this spectral sequence. Indeed, except for the fact that he replaced homology by cohomology here, this was the approach used in [Mil] in proving Theorem 3.1. Wall’s papers [Wall3] and [Wall6] on the Steenrod algebra may be viewed as being motivated by this approach. In fact one can show that the Steenrod comodule structure of H_{∗}(MSO; ℤ/p) is relatively simple. For p odd, A_{∗}^{p} is the tensor product of an exterior algebra and a polynomial algebra, and the polynomial part is a Hopf subalgebra. It turns out ([Swit], Lemma 20.38) that H_{∗}(MSO; ℤ/p) is a free comodule on the polynomial part of A_{∗}^{p}, and thus the Adams spectral sequence is fairly easy to analyze. For p = 2, A_{∗}^{2} ≅ (ℤ/2)[ξ_{1}, ξ_{2}, …] with ξ_{i} of degree 2^{i} − 1 and with coproduct

Δ(ξ_{k}) = ∑_{i+j=k} ξ_{j}^{2^{i}} ⊗ ξ_{i},

and H_{∗}(MSO; ℤ/2) turns out to be the direct sum of a free A_{∗}^{2}-comodule and a free comodule over (A^{∗}/A^{∗}Sq^{1})^{∗}; this fact is basically equivalent to one of the main results in [Wall5], for which Wall gave a simpler proof in [Wall6]. Another proof is given in [Peng2], and the algebra structure instead of the comodule structure is computed in [Pap1] and [Peng1]. Once again, the Adams spectral sequence is computable (see [Swit], pages 510–516 for details). One can obtain in this way another proof of Wall’s Theorem 3.2, with the drawback that the geometry of (1) is submerged. The compensation is that the Hurewicz map π_{∗}(MSO) → H_{∗}(MSO; ℤ/p) can be read off directly, giving a clearer explanation of the second part of Wall’s Theorem (the fact that classes in Ω are determined by Pontrjagin and Stiefel-Whitney numbers). A few more variants of Wall’s proof have been given, most notably:

1. a very quick and slick proof by Taylor [Tayl] of Theorem 3.2, which uses the Atiyah-Hirzebruch spectral sequence to study the Hurewicz maps for MSO with ℤ/2 and ℤ/4 coefficients;

2. a proof of Gray [Gray], which uses a purely algebraic description of 𝔚, together with multiplicative structure of the Atiyah-Hirzebruch spectral sequence; and

3. a proof of Papastavridis [Pap2], a variant on the Adams spectral sequence approach.

4 Cobordism theory since Wall’s work

Wall’s paper [Wall2] on cobordism contains the words:

In fact, all properties of Ω are as simple as they could possibly be (if my results should strike the reader as complicated, let him try and work out cobordism theory for the spinor group).

The implicit challenge here was to compute Ω^{spin}, the cobordism ring of oriented manifolds equipped with spin structures (or equivalently, lifts of the stable normal bundle M → BSO to a map M → BSpin, where Spin(n) is the double cover of SO(n)). This challenge was eventually met by Anderson, Brown, and Peterson [AnBrP], and indeed the answer is quite complicated. However, there are a few similarities to Wall’s work on Ω, which certainly served as an important model. Again, all torsion is of order 2, and again, the ℤ/2-homology of the relevant Thom spectrum MSpin is a direct sum of relatively simple A_{∗}^{2}-comodules of only a few types. The difference between MSpin and MSO at the prime 2 turns out to involve, as one would expect, primarily Sq^{2}. More precisely, A(1)_{∗} = (ℤ/2)[ξ_{1}, ξ_{2}, …]/(ξ_{1}^{4}, ξ_{2}^{2}, …) is dual to the subalgebra of A_{2}^{∗} generated by Sq^{1} and Sq^{2}, and H_{∗}(MSpin; ℤ/2) turns out to be a direct sum of A_{∗}^{2}-comodules of the form A_{∗}□_{A(1)_{∗}}M, with M one of three A(1)_{∗}- comodules, just as if A(0)_{∗} is dual to the subalgebra of A_{2}^{∗} generated by Sq^{1}, then H_{∗}(MSO; ℤ/2) turns out to be a direct sum of A_{∗}^{2}-comodules of the form A_{∗}□_{A(0)_{∗}}M, with M one of two A(0)_{∗}-comodules. In fact, the [AnBrP] proof of this relies quite heavily on the technique developed by Wall in [Wall6]. This technology can be carried quite far (see for example [Stolz]), and has applications which will be discussed elsewhere in this volume, especially in the chapters [RS] on positive scalar curvature and [Tms] on elliptic cohomology.

While it would take us quite far afield to discuss everything that’s happened in cobordism theory since Wall’s fundamental papers, much of which can be found in the book [Stong] or the survey [Land1], we should mention what are perhaps the most important trends. They stem from remarkable properties of the spectrum MU for complex cobordism Ω^{U} (the cobordism theory for manifolds with an almost complex structure on the stable normal bundle), which are described in detail in the books [Ad2] and [Rav]. It turns out not only that Ω^{U} is a polynomial ring over ℤ [Mil], but also that this ring carries a natural “formal group” structure coming from the Conner-Floyd Chern classes. Quillen [Quil] proved that the formal group law on Ω^{U} is “universal,” i.e., that it can be mapped to every other (onedimensional commutative) formal group law. From Quillen’s theorem one can deduce (see for instance [Rav], Chapter 4, or [Ad2], Part II, §§15–16) that when localized at a prime p, MU splits as a sum of suspensions of a very special spectrum BP, which plays a very important role in modern stable homotopy theory. For example, one obtains from it the “Morava Ktheory” spectra K(n), which are periodic of periods 2(p^{n} −1). When p = 2 and n = 1, K(n) is just ordinary complex K-theory with ℤ/2 coefficients, but the other cases have no such simple geometric interpretations. This fact is related (see [Rav], p. 135) to the Conner-Floyd Theorem [CoF] that complex K-theory may be simply constructed out of complex cobordism as

K_{∗}(X) ≅ Ω_{∗}^{U}(X) ⊗_{Ω_{∗}^{U}} K_{∗}(pt).  (2)

There is no direct link between all this and Wall’s work on cobordism, though it does turn out that MSO when localized at an odd prime splits as a sum of suspensions of BP, even though MSO localized at 2 looks quite different and is a direct sum of (shifted) Eilenberg-MacLane spectra for the groups ℤ_{(2)} and ℤ/2. As pointed out in [TayW], this is immediately related to the structure of the L-spectra which are discussed elsewhere in this volume, which are KO-module spectra at odd primes and Eilenberg-MacLane at 2. The reason is that L^{•}(ℤ) is an MSO-module spectrum, and all other L-spectra are module spectra over L^{•}(ℤ) and hence also over MSO.

A related development is the recent interest in “elliptic homology” theories, and their connection with cobordism. To explain something about these one first needs the idea of a “genus,” which in its most general formulation is simply a ring homomorphism from some cobordism ring to some standard commutative ring Λ (such as ℚ). The important classical examples are the mod 2 Euler characteristic 𝔑 → ℤ/2, the signature Ω → ℤ for oriented manifolds, the Todd genus Ω^{U} → ℤ for complex manifolds, and the Â-genus Ω^{spin} → ℤ. Under favorable circumstances, a genus is really the induced map on homotopy groups of some map of ring spectra. For example, the K-orientation of complex manifolds gives a map of ring spectra MU → K, which when composed with the Chern character to ordinary homology, gives rise on taking homotopy groups to the Todd genus, and the KO-orientation of spin manifolds gives a map of ring spectra MSpin → KO which, when composed with the Pontrjagin character to ordinary homology, gives rise on taking homotopy groups to the Â-genus.

Now if Λ is a ℚ-algebra and φ : Ω → Λ is a genus, Hirzebruch associated to φ the formal power series

g(x) = ∑_{n=0}^{∞} \frac{1}{2n+1}φ(ℂℙ^{2n})x^{2n+1}.

It was discovered by Ochanine [Och1] that the genus φ vanishes on all manifolds of the form ℂℙ(ξ) with ξ an even-dimensional complex vector bundle over a closed oriented manifold if and only if g(x) is an elliptic integral

g(x) = ∫_{0}^{x} \frac{1}{\sqrt{1 − 2δz^{2} + εz^{4}}} dz.  (3)

In this case φ is called an elliptic genus. (For expositions of all this, see the papers in [Land2], especially Landweber’s survey on pages 55–68, or the book [HiBeJ], as well as [Tms].) This suggested the search for multiplicative homology theories, related to cobordism spectra, whose coefficient rings are rings of automorphic forms (of which elliptic integrals are a special case). The first result in this direction was the discovery by Landweber, Ravenel, and Stong [LaRS] of a multiplicative homology theory at odd primes, having coefficient ring M = ℤ[\frac{1}{2}][δ, ε]. They showed the elliptic genus associated to (3) gives a ring homomorphism Ω → M, and that while X ↦ Ω_{∗}(X)⊗_{Ω} M is not a homology theory, it becomes a homology theory after inverting ε, δ^{2} − ε, or the discriminant Δ = ε(δ^{2} − ε)^{2}. (The proof relies on the fact that at odd primes, MSO closely resembles MU, together with universality of the formal group law on MU.) For another proof and related results, see also [Fr], [Bryl], [Bak], and [Tms]. More dramatic are recent elegant results of [KrSt] and [Hov], which construct integral homology theories El_{∗} out of spin cobordism, which agree with various versions of the Landweber, Ravenel, and Stong theory at odd primes. For the work of Kreck and Stolz, an understanding of the very delicate structure of H_{∗}(MSpin; ℤ/2) as a comodule over A_{∗}^{2} is the key. This structure gives rise to theorems of Conner-Floyd type analogous to (2):

El_{∗}(X) ≅ Ω_{∗}^{spin}(X) ⊗_{Ω_{∗}^{spin}} El_{∗}(pt),
KO_{∗}(X) ≅ El_{∗}(X) ⊗_{El_{∗}(pt)} KO_{∗}(pt)

([HoHo], [Hov]). As we’ve seen, the roots of our understanding of this may be traced back to Wall’s work on cobordism.

References

[Ad1] J. F. Adams, Exposé sur les travaux de C. T. C. Wall sur l’algèbre de cobordisme Ω, Bull. Soc. Math. France 87 (1959), 281–284.

[Ad2] J. F. Adams, Stable Homotopy and Generalised Homology, Chicago Lectures in Math., vol. 17, Univ. of Chicago Press, Chicago, 1974; reprinted, 1995.

[AnBrP] D. W. Anderson, E. H. Brown, Jr., and F. P. Peterson, The structure of the spin cobordism ring, Ann. of Math. 86 (1967), 271–298.

[Atiyah] M. F. Atiyah, Bordism and cobordism, Proc. Camb. Phil. Soc. 57 (1961), 200–208.

[Bak] A. Baker, Elliptic genera of level N and elliptic cohomology, J. London Math. Soc. (2) 49 (1994), 581–593.

[Bryl] J.-L. Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology 29 (1990), 461–480.

[CoF] P. E. Conner and E. E. Floyd, The relation of cobordism to Ktheories, Lecture Notes in Math., vol. 28 , Springer-Verlag, Berlin and New York, 1966.

[Dold] A. Dold, Erzeugende der Thomschen Algebra 𝔑_{∗}, Math. Z. 65 (1956), 25–35.

[Gray] B. Gray, Products in the Atiyah-Hirzebruch spectral sequence and the calculation of MSO_{∗}, Trans. Amer. Math. Soc. 260 (1980), 475–483.

[Fr] J. Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992), 43–65.

[HiBeJ] F. Hirzebruch, T. Berger, and R. Jung, Manifolds and modular forms (with appendices by N.-P. Skoruppa and by P. Baum), Aspects of Math., vol. 20, Vieweg, Braunschweig, 1992.

[HoHo] M. J. Hopkins and M. A. Hovey, Spin cobordism determines real K-theory, Math. Z. 210 (1992), 181–196.

[Hov] M. A. Hovey, Spin bordism and elliptic homology, Math. Z. 219 (1995), 163–170.

[KrSt] M. Kreck and S. Stolz, ℍℙ^{2}-bundles and elliptic homology, Acta Math. 171 (1993), 231–261.

[Land1] P. S. Landweber, A survey of bordism and cobordism, Math. Proc. Cambridge Philos. Soc. 100 (1986), 207–223.

[Land2] P. S. Landweber (editor), Elliptic curves and modular forms in algebraic topology, Proceedings, Princeton, 1986, Lecture Notes in Math., vol. 1326, Springer-Verlag, Berlin and New York, 1988.

[LaRS] P. S. Landweber, D. C. Ravenel, and R. E. Stong, Periodic cohomology theories defined by elliptic curves, in The Čech centennial (Boston, 1993), Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 317–337.

[Mil] J. W. Milnor, On the cobordism ring and a complex analogue, Part I, Amer. J. Math. 82 (1960), 505–521.

[Och1] S. Ochanine, Sur les genres multiplicatifs définis par des intégrales elliptiques, Topology 26 (1987), 143–151.

[Pap1] S. Papastavridis, An algebraic proof of a theorem of D. Pengelley on the structure of H_{∗}(MSO; ℤ/2), in Current trends in algebraic topology, Part 1 (London, Ont., 1981), CMS Conf. Proc., 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 507–510.

[Pap2] S. Papastavridis, Generators of H^{∗}(MSO; ℤ/2) as a module over the Steenrod algebra, and the oriented cobordism ring, Proc. Amer. Math. Soc. 84 (1982), 285–290.

[Peng1] D. J. Pengelley, The A-algebra structure of Thom spectra: MSO as an example, in Current trends in algebraic topology, Part 1 (London, Ont., 1981), CMS Conf. Proc., 2, Amer. Math. Soc., Providence, RI, 1982, pp. 511–513.

[Peng2] D. J. Pengelley, The mod two homology of MSO and MSU as A comodule algebras, and the cobordism ring, J. London Math. Soc. 25 (2) (1982), 467–472.

[Quil] D. G. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293– 1298.

[Ran1] A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery, Mathematical Notes, vol. 26, Princeton Univ. Press, Princeton, NJ, 1981.

[Ran2] A. Ranicki, Algebraic L-theory and topological manifolds, Cambridge Tracts in Mathematics, vol. 102, Cambridge Univ. Press, Cambridge, 1992.

[Rav] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Pure and Applied Math., vol. 121, Academic Press, Orlando and London, 1986.

[RS] J. Rosenberg and S. Stolz, Metrics of positive scalar curvature and connections with surgery, this volume.

[Stolz] S. Stolz, Splitting certain MSpin-module spectra, Topology 33 (1994), 159–180.

[Stong] R. Stong, Notes on Cobordism Theory, Mathematical Notes, vol. 7, Princeton Univ. Press, Princeton, NJ, 1968.

[Swit] R. M. Switzer, Algebraic Topology—Homotopy and Homology, Grundlehren der Math. Wissenschaften, vol. 212, Springer-Verlag, Berlin and New York, 1975.

[Tayl] L. R. Taylor, 2-local cobordism theories, J. London Math. Soc. (2) 14 (1976), 303–308.

[TayW] L. R. Taylor and B. Williams, Surgery spaces: formulæ and structure, in Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978), pp. 170–195, Lecture Notes in Math., vol. 741, Springer, Berlin, 1979.

[Thom] R. Thom, Quelques propriétés globales des variétés différentiables, Comm. Math. Helv. 28 (1954), 17–86.

[Tms] C. B. Thomas, Elliptic cohomology, this volume.

[Wall1] Note on the cobordism ring, Bull. Amer. Math. Soc. 65 (1959), 329–331.

[Wall2] Determination of the cobordism ring, Ann. of Math. 72 (1960), 292–311.

[Wall3] Generators and relations for the Steenrod algebra, Ann. of Math. 72 (1960), 429–444.

[Wall4] Cobordism of pairs, Comm. Math. Helv. 35 (1961), 136–145.

[Wall5] Cobordism exact sequences for differential and combinatorial manifolds, Ann. of Math. 77 (1963), 1–15.

[Wall6] A characterisation of simple modules over the Steenrod algebra mod 2, Topology 1 (1962) 249–254.