SPIN^{c}-STRUCTURES AND HOMOTOPY EQUIVALENCES

Robert E. Gompf

Abstract. We show that a homotopy equivalence between manifolds induces a correspondence between their spin^{c}-structures, even in the presence of 2-torsion. This is proved by generalizing spin^{c}-structures to Poincaré complexes. A procedure is given for explicitly computing the correspondence under reasonable hypotheses.

1. Introduction

The theory of spin^{c}-structures has attained new importance through its recent application to the topology of smooth 4-manifolds. Among smooth, closed, oriented 4-manifolds (with b_{1} + b_{+} odd) a typical homeomorphism type contains many diffeomorphism types. The only invariants known to distinguish such diffeomorphism types are those arising from gauge theory, as pioneered by Donaldson (e.g., [DK]). The most efficient approach currently known is to assign a Seiberg-Witten invariant (e.g., [Mo]) to any such 4-manifold X with a fixed spin^{c}-structure. To extract the most information from these invariants, one must understand how spin^{c}-structures transform under homeomorphisms. This is straightforward if H^{2}(X; ℤ) has no 2-torsion (for example, if X is simply connected), for then the Chern class will distinguish any two spin^{c}-structures on X. The general case is less obvious, however. In high dimensions, a homeomorphism between smooth manifolds need not be covered by an isomorphism of their tangent bundles. While such isomorphisms always exist in dimension 4, they are not canonical, and automorphisms of the tangent bundle covering id_{X} may permute the spin^{c}-structures on X. (For example, such an automorphism over ℝP^{3} or ℝP^{3} × S^{1} can be constructed from the diffeomorphism ℝP^{3} → SO(3).) In this note, we show how to canonically assign to any orientation-preserving proper homotopy equivalence X_{1} → X_{2} between manifolds a correspondence between spin^{c}-structures on X_{1} and those on X_{2}.

Our approach is to generalize the theory of spin and spin^{c}-structures from SO(n) to more general structure groups H. Most of the homotopy of SO(n) does not enter into the theory. In fact, it suffices for H to be path connected with a nontrivial double cover so that we can generalize the definition spin^{c}(n) = (spin(n)×spin(2))/ℤ_{2}.

Typeset by AMS-TEX

Date: May 2, 1997.

Partially supported by NSF grant DMS-9625654. Research at MSRI is supported in part by NSF grant DMS-9022140.

The resulting theory generalizes the classical theory in the obvious way, for example, with spin^{c}-structures on a bundle ξ over X classified by H^{2}(X; ℤ) whenever W_{3}(ξ) = 0 (Proposition 1). Ultimately, the map BSO → BSG of classifying spaces allows us to generalize spin^{c}-structures from smooth manifolds to Poincaré complexes, and the latter theory has the required functoriality with respect to homotopy equivalences by naturality of the Spivak normal fibration (Theorem 5). Under reasonable hypotheses, one can explicitly compute the correspondence of spin^{c}-structures induced by a homotopy equivalence; a procedure is given following Theorem 5. The concluding remarks include other characterizations of classical spin^{c}-structures.

The author wishes to acknowledge Larry Taylor for helpful discussions, and the Mathematical Sciences Research Institute for their hospitality.

2. Generalized spin^{c}-structures

A naive approach to generalizing the theory of spin and spin^{c}-structures would be to define spin(H) to be a preassigned double cover of a path connected topological group H, and let spin^{c}(H) denote the group spin(H × SO(2)) diagonally double covering H × SO(2). One could then generalize the theory in the obvious way, using principal spin(H) and spin^{c}(H)-bundles, the natural epimorphisms from spin^{c}(H) to H and SO(2), and the involution of spin^{c}(H) induced by conjugation on SO(2) = U(1). However, to avoid the difficulties of adapting principal bundle theory to spherical fibrations, we translate the argument into the language of classifying spaces, replacing epimorphisms of groups with kernel ℤ_{2} or SO(2) by fibrations of the corresponding classifying spaces with fiber Bℤ_{2} = K(ℤ_{2}, 1) = ℝP^{∞} or BSO(2) = K(ℤ, 2) = ℂP^{∞}, respectively. We remove the groups from the theory while keeping the suggestive notation, obtaining a theory of spin and spin^{c}-structures on bundles or fibrations classified by a universal bundle (fibration) ξ_{H} → BH, where BH is homotopy equivalent to a simply connected CW -complex, and a nonzero class w ∈ H^{2}(BH; ℤ_{2}) is specified (corresponding to a choice of double cover of H). We can recover the classical theory by setting BH = BSO(n) (n ≥ 2), with w the unique nonzero class w^{∗} ∈ H^{2}(BSO(n); ℤ_{2}) ≅ ℤ_{2}.

Recall [W] that any map f : X → Y can be transformed into a fibration by replacing X by the space P of paths from X to Y in the mapping cylinder of f. The initial point fibration p_{0} : P → X has contractible fiber, and the endpoint fibration p_{1} : P → Y is homotopic to f ∘ p_{0}. The fiber F of p_{1} is homotopy equivalent to a CW -complex if X and Y are [M1], and p_{0}|F is a fibration with fiber the loop space ΩY.

Now let (BH, w) be as above. Then w defines epimorphisms H_{2}(BH; ℤ_{2}) → ℤ_{2} and hence φ_{w} : π_{2}(BH) → ℤ_{2}. We apply the previous paragraph to the map BH → K(ℤ_{2}, 2) induced by φ_{w}, and let Bspin(H, w) denote the fiber F. The fibration Bspin(H, w) → BH induces isomorphisms of π_{i}(Bspin(H, w)) with ker φ_{w} for i = 2 and π_{i}(BH) otherwise, and its fiber is K(ℤ_{2}, 1) = ℝP^{∞}. Now we define Bspin^{c}(H, w) to be Bspin(H^{×}, w + w^{∗}), where BH^{×} = BH × BSO(2). We immediately obtain fibrations p_{H} and p_{SO(2)} of Bspin^{c}(H, w) over BH and BSO(2), with fibers Bspin(SO(2), w^{∗}) = K(ℤ, 2) and Bspin(H, w), respectively, and each fibration restricted to the opposite fiber is the map arising from the definition of Bspin(·). (Compare with the projections of spin^{c}(H, w) to H and SO(2) on the level of groups.) By obstruction theory, complex conjugation on the second factor BSO(2) = ℂP^{∞} of BH^{×} lifts uniquely from BH^{×} to a map on Bspin^{c}(H, w) whose square is fiber homotopic to the identity, and the map is homotopic to conjugation on each ℂP^{∞}-fiber of p_{H}.

To define spin^{c}-structures over H, recall that an H-bundle (or fibration) ξ → X over a CW -complex is classified by a bundle map

\begin{matrix}ξ & ⟶^{f̃} & ξ_{H} \\ ↓ & & ↓ \\ X & ⟶^{f} & BH \end{matrix} .

For two choices of classifying map f̃, there is a canonical homotopy (up to homotopy rel 0,1) between the corresponding maps f, characterized by lifting to a homotopy of the maps f̃ through bundle maps. This allows us to define spin^{c}-structures in a manner independent of the choice of f̃.

Definition. A spin structure on an H-bundle (fibration) ξ (relative to w) is a function assigning to each classifying bundle map f̃ : ξ → ξ_{H} a homotopy class of lifts f̂: X → Bspin(H, w) of f : X → BH, such that for two choices of f̃ the canonical homotopy between the maps f lifts to a homotopy of the corresponding maps f̂. A spin^{c}-structure is defined similarly with spin replaced by spin^{c}.

We denote the sets of spin and spin^{c}-structures on an H-bundle ξ by S(ξ, w) and S^{c}(ξ, w), respectively. Note that in either case, any lift of a single f with a specified f̃ uniquely determines such a structure, but changing f̃ with f fixed may result in an automorphism of S(ξ, w) or S^{c}(ξ, w).

To define characteristic classes, let Y ⊂ X be a possibly empty subcomplex, and let τ be a trivialization of ξ|Y. Then we can assume that the classifying map f : X → BH of ξ is constant on Y, and that τ determines the restriction f̃|Y : ξ|Y → ξ_{H}. Set w_{2}(ξ, τ) = f^{∗}(w) ∈ H^{2}(X, Y; ℤ_{2}) and W_{3}(ξ, τ) = βw_{2}(ξ, τ) ∈ H^{3}(X, Y; ℤ), where β is the Bockstein homomorphism. Any spin^{c}-structure s ∈ S^{c}(ξ, w) determines a homotopy class of lifts f̂: X → Bspin^{c}(H, w) of f, and we define a trivialization τ̂ of s|Y over τ to be a choice of f̂ (within the given homotopy class) that is constant on Y, up to homotopies through such maps. (Equivalently, τ̂ is a spin^{c}-structure on X/Y that pulls back to s on X.) We define Chern classes by setting c_{1}(s, τ̂) = f̂^{∗}p_{SO(2)}^{∗}(c) ∈ H^{2}(X, Y; ℤ), where c ∈ H^{2}(BSO(2), ℤ) ≅ ℤ is the generator c_{1}(ξ_{SO(2)}). If Y is empty, we use the notation w_{2}(ξ), W_{3}(ξ), c_{1}(s).

Proposition 1. The set S(ξ, w) of spin structures on an H-bundle (or fibration) ξ → X is nonempty if and only if w_{2}(ξ) = 0. If so, then H^{1}(X; ℤ_{2}) acts freely and transitively on S(ξ, w). The set S^{c}(ξ, w) is nonempty if and only if W_{3}(ξ) = 0, and if so, then H^{2}(X; ℤ) acts freely and transitively on it. For s ∈ S^{c}(ξ, w) and a ∈ H^{2}(X; ℤ), we have c_{1}(s+a) = c_{1}(s)+2a. Conjugation induces an involution on S^{c}(ξ, w) that reverses signs of Chern classes and the H^{2}(X; ℤ)-action. For Y ⊂ X and τ̂ as above, c_{1}(s, τ̂) reduces modulo 2 to w_{2}(ξ, τ).

Thus, choosing a base point in S(ξ, w) or S^{c}(ξ, w) (if nonempty) identifies it with H^{1}(X; ℤ_{2}) or H^{2}(X; ℤ).

Proof. The first two sentences are immediate from obstruction theory, since the fiber of Bspin(H, w) → BH is K(ℤ_{2}, 1). In fact, w_{2}(ξ, τ) is the obstruction to lifting f to a map f̂: X → Bspin(H, w) with f̂|Y constant, as can be seen by first considering the case where Y contains the 1-skeleton of X. Similarly, H^{2}(X; ℤ) acts as required on S^{c}(ξ, w) (when nonempty) via difference classes, since the fiber of p_{H} is K(ℤ, 2). Now recall that Bspin^{c}(H, w) = Bspin(H^{×}, w+w^{∗}) with BH^{×} = BH × BSO(2). Thus, a lift of f to f̂: X → Bspin^{c}(H, w) with f̂|Y constant is the same as a choice of complex line bundle L → X with a trivialization τ_{L} over Y, together with a spin structure on the bundle ξ × L → X (classified by BH × BSO(2)) whose defining lift f̂ is constant on Y. The resulting spin^{c}-structure s with trivialization τ̂ over τ will satisfy c_{1}(s, τ̂) = c_{1}(L, τ_{L}), since p_{SO(2)} ∘ f̂ is the classifying map of L. Such a structure exists if and only if 0 = w_{2}(ξ ×L, τ ×τ_{L}) = w_{2}(ξ, τ)+w_{2}(L, τ_{L}), or equivalently w_{2}(ξ, τ) = w_{2}(L, τ_{L}) = c_{1}(L, τ_{L})|_{2}. Thus, S^{c}(ξ, w) is nonempty if and only if w_{2}(ξ) has a lift to H^{2}(X; ℤ), i.e., W_{3}(ξ) = 0, and any c_{1}(s, τ̂) reduces mod 2 to w_{2}(ξ, τ). Given s, s′ ∈ S^{c}(ξ, w), the difference class d(s, s′) takes coefficients in π_{2}(K(ℤ, 2)), where K(ℤ, 2) is the fiber of p_{H}. Since (p_{SO(2)})_{∗} : π_{2}(K(ℤ, 2)) → π_{2}(BSO(2)) is multiplication by 2, we have 2d(s, s′) = c_{1}(s′) − c_{1}(s). Equivalently, c_{1}(s + a) = c_{1}(s) + 2a for a = d(s, s′). The assertion about conjugation is clear from the way it lifts to Bspin^{c}(H, w). □

Now suppose we are given pairs (BH, w) and (BH′, w′) as before, and a map h : BH → BH′ covered by a bundle map h̃ : ξ_{H} → ξ_{H′}, with h^{∗}w′ = w. Then any H-bundle ξ → X determines an H′-bundle ξ′ → X with the same w_{2} and W_{3}, and h determines maps Bspin(H, w) → Bspin(H′, w′) and Bspin^{c}(H, w) → Bspin^{c}(H′, w′). We obtain canonical equivariant identifications S(ξ, w) ≅ S(ξ′, w′) and S^{c}(ξ, w) ≅ S^{c}(ξ′, w′), and the latter preserves Chern classes and conjugation. On the other hand, given an H-bundle map g̃ : ξ_{1} → ξ_{2} covering g : X_{1} → X_{2}, we have induced maps g^{∗} : S(ξ_{2}, w) → S(ξ_{1}, w) and g^{∗} : S^{c}(ξ_{2}, w) → S^{c}(ξ_{1}, w) that are equivariantly equivalent to g^{∗} : H^{1}(X_{2}; ℤ_{2}) → H^{1}(X_{1}, ℤ_{2}) and g^{∗} : H^{2}(X_{2}; ℤ) → H^{2}(X_{1}; ℤ) when the domains are nonempty, and characteristic classes and conjugation are preserved in the obvious way. If g is a homotopy equivalence, then the maps g^{∗} are isomorphisms.

Examples 2. (a) If h : BSO(m) → BSO(n), 2 ≤ m ≤ n, is induced by the usual inclusion of groups, we recover the stabilization-invariance of classical spin and spin^{c}-structures. We are free to pass to the limiting group SO, eliminating the dependence on n.

(b) An oriented topological n-manifold X has the homotopy type of a CW - complex, and it has a tangent bundle classified by a map into the universal bundle over BST OP (n) (eg. [KS]). There is a canonical map h : BSO(n) → BST OP (n) that corresponds to interpreting ξ_{SO(n)} as a topological bundle and is a π_{2}-iso-morphism of simply connected spaces. We immediately obtain a theory of spin and spin^{c}-structures on oriented topological manifolds by using their tangent bundles (stabilized if n < 2). As before, the theory is stabilization-invariant, and we can pass to the limiting case of BST OP. On smooth manifolds, the new theory canonically reduces via h to the classical theory. However, any orientation-preserving homeomorphism g : X_{1} → X_{2} induces an isomorphism of topological tangent bundles, hence, isomorphisms g^{∗} : S(X_{2}) ≅ S(X_{1}) and g^{∗} : S^{c}(X_{2}) ≅ S^{c}(X_{1}) as above.

To generalize to homotopy equivalences, we need one further construction. Suppose we are given a bundle map

ξ_{H} × ξ_{H′}  ⟶^{k̃} ξ_{H′′}
↓  ↓
BH × BH′ ⟶^{k} BH′′

with k^{∗}(w′′) = w + w′. Then a pair of bundles ξ, ξ′ → X classified by BH, BH′ determine an H′′-bundle ξ′′ → X, and w_{2} and W_{3} add.

Proposition 3. A trivialization of ξ′′ induces equivariant isomorphisms

k_{∗} : S(ξ, w) → S(ξ′, w′) and k_{∗} : S^{c}(ξ, w) → S^{c}(ξ′, w′),

and the latter preserves conjugation and Chern classes.

Proof. By obstruction theory, the map k uniquely determines a map k̂ making the diagram

Bspin(H, w) × Bspin(H′, w′) ⟶^{k̂} Bspin(H′′, w′′)
↓p_{1}  ↓p_{2}
BH × BH′  ⟶^{k}  BH′′

commute, and a similar diagram is induced for spin^{c} via the map k × k_{0}, where k_{0} : BSO(2)×BSO(2) → BSO(2) induces addition on π_{2}. The diagrams determine a map k_{#} : S(ξ, w) × S(ξ′, w′) → S(ξ′′, w′′) and similarly for S^{c}. In the latter case, k_{#} commutes with conjugation and adds Chern classes. In either case, k̂ restricts to addition on the homotopy groups of the fibers of p_{1} and p_{2}, so difference classes add under k_{#}, and for suitably chosen base points k_{#} is given by addition on H^{1}(X; ℤ_{2}) or H^{2}(X; ℤ) whenever its domain is nonempty. Now a trivialization of ξ′′ determines a trivial spin^{c}-structure s′′ ∈ S^{c}(ξ′′, w′′). Since W_{3}(ξ) + W_{3}(ξ′) = W_{3}(ξ′′) = 0, it follows that S^{c}(ξ, w) is nonempty if and only if S^{c}(ξ′, w′) is. For each s ∈ S^{c}(ξ, w) there is a unique “inverse” s′ ∈ S^{c}(ξ′, w′) with k_{#}(s, s′) = s′′. Let k_{∗}(s) equal the conjugate of s′. Then k_{∗} : S^{c}(ξ, w) → S^{c}(ξ′, w′) is an equivariant isomorphism, and it preserves conjugation and Chern classes since s′′ is conjugationinvariant with c_{1}(s′′) = 0. A similar procedure (with k_{∗}(s) = s′) works for spin structures. □

Example 4. Any oriented, smooth n-manifold X admits a unique isotopy class of proper embeddings in ℝ^{N} for N sufficiently large. This determines a normal bundle νX that is unique up to stabilization. Since the tangent bundle τX satisfies τX ⊕ νX = τℝ^{N}|X and the latter bundle is canonically trivial, the obvious map BSO(n)×BSO(N−n) → BSO(N) determines canonical equivariant identifications S(νX, w^{∗}) ≅ S(τX, w^{∗}) and S^{c}(νX, w^{∗}) ≅ S^{c}(τX, w^{∗}), the latter preserving Chern classes and conjugation.

Theorem 5. Let (X, ∂X) be an oriented, possibly noncompact Poincaré pair. There is a canonical procedure for defining sets S(X) and S^{c}(X) of spin and spin^{c}- structures on X having the structure described in Proposition 1 (with respect to the usual classes w_{2}(X) and W_{3}(X)). For (X, ∂X) a smooth manifold, the theory is canonically equivariantly equivalent to the standard one (preserving Chern classes and conjugation). For pairs (X_{i}, ∂X_{i}) as above, any orientation-preserving, pairwise, proper homotopy equivalence g : (X_{1}, ∂X_{1}) → (X_{2}, ∂X_{2}) induces equivariant isomorphisms g^{∗} : S(X_{2}) ≅ S(X_{1}) and g^{∗} : S^{c}(X_{2}) ≅ S^{c}(X_{1}), the latter preserving Chern classes and conjugation, and the construction is functorial for such maps g.

Proof. The pair (X, ∂X) has a canonical Spivak normal fibration [S] defined by embedding (X, ∂X) pairwise and properly in half-space ℝ^{N} ×([0, ∞), {0}) (uniquely for N sufficiently large), and making a fibration out of the collapsing map of the boundary of a regular neighborhood. The resulting oriented spherical fibration over X is classified by a fiber-preserving map into the universal spherical fibration, whose base space stabilizes to BSG. As in Example 2(b), there is a canonical map h : BSO → BSG induced by the spherical fibrations ξ_{SO(n)} − (0-section), and h is a π_{2}-isomorphism of simply connected spaces. We immediately obtain S(X), S^{c}(X) and characteristic classes satisfying Proposition 1, using the Spivak fibration and BSG. (The resulting classes w_{2}(X) and w_{3}(X) are well known.) For (X, ∂X) a smooth manifold, the theory is canonically equivalent (via h) to that of the stable normal bundle, which is the usual theory over the tangent bundle by Example 4. A homotopy equivalence g as above induces a fiber-preserving map of the corresponding Spivak fibrations, and hence, the required maps g^{∗}. □

The map g^{∗} : S^{c}(X_{2}) ≅ S^{c}(X_{1}) induced by a homotopy equivalence can frequently be computed explicitly. We consider the case where X_{2} contains a 1dimensional subcomplex with a regular neighborhood N_{2} that is a manifold, such that H^{2}(X_{2}, N_{2}; ℤ) has no 2-torsion. We also assume that g : X_{1} → X_{2} restricts to a homeomorphism from N_{1} = g^{−1}(N_{2}) to N_{2}. These conditions are always satisfied if g is a homeomorphism between smooth manifolds, for example by taking N_{2} to be a neighborhood of the 1-skeleton of X_{2}. Now the map g^{∗} : H^{∗}(X_{2}, N_{2}) ≅ H^{∗}(X_{1}, N_{1}) is an isomorphism. A (stable) trivialization τ_{2} of the tangent bundle of N_{2} (or equivalently, of the stable normal bundle) pulls back via g|N_{1} to a trivialization τ_{1} over N_{1}, and g^{∗}w_{2}(X_{2}, τ_{2}) = w_{2}(X_{1}, τ_{1}). Given spin^{c}-structures s_{i} ∈ S^{c}(X_{i}), pick any trivializations τ̂_{i} of s_{i}|N_{i} over τ_{i}. Then by Proposition 1, g^{∗}c_{1}(s_{2}, τ̂_{2}) − c_{1}(s_{1}, τ̂_{1}) reduces to zero mod 2. Since H^{2}(X_{1}, N_{1}; ℤ) has no 2-torsion, there is a unique class δ(s_{1}, s_{2}) ∈ H^{2}(X_{1}, N_{1}; ℤ) with 2δ(s_{1}, s_{2}) = g^{∗}c_{1}(s_{2}, τ̂_{2}) − c_{1}(s_{1}, τ̂_{1}). If we change τ̂_{i} with τ_{i} fixed, then δ(s_{1}, s_{2}) changes by the coboundary of a cochain in N_{1}, so it represents a class d(s_{1}, s_{2}) ∈ H^{2}(X_{1}; ℤ) that depends only on s_{1} and s_{2} (τ_{i} fixed). But δ(s_{1}, s_{2}) vanishes for s_{1} = g^{∗}s_{2} and τ̂_{1} given by pulling back τ̂_{2}, and a change of s_{i} changes 2δ(s_{1}, s_{2}) by twice the corresponding relative difference class (by the addition formula of Proposition 1 applied to X_{i}/N_{i}). Thus, d(s_{1}, s_{2}) is precisely the difference class d(s_{1}, g^{∗}s_{2}), in a form accessible to computation.

Remarks. (a) Spin^{c}-structures have several other convenient characterizations. As we observed in proving Proposition 1, a spin^{c}-structure on ξ → X is the same as a line bundle L and spin structure on ξ × L → X. For a different approach, recall that Milnor [M2] observed that a spin structure on an oriented vector bundle over a CW -complex is equivalent (after stabilizing if necessary) to a trivialization over the 1-skeleton that can be extended over the 2-skeleton, just as an orientation is a trivialization over the 0-skeleton that extends over the 1-skeleton. Similarly, a spin^{c}-structure over an oriented vector bundle is equivalent (after stabilizing if the fiber dimension is odd or ≤ 2) to a complex structure over the 2-skeleton that can be extended over the 3-skeleton. To see this, observe that the map of classifying spaces induced by inclusion i : U(n) → SO(2n) lifts canonically to a map j : BU(n) → Bspin^{c}(SO(2n), w^{∗}) by first lifting the map id × B det : BU(n) → BU(n)×BSO(2) to Bspin^{c}(U(n), i^{∗}w^{∗}). (In fact, the corresponding diagram exists on the group level.) Thus, any complex structure determines a spin^{c}-structure (and the correspondence preserves c_{1} and conjugation). For n ≥ 2, this correspondence is bijective for 2-complexes and surjective for 3-complexes, since the map j has a 2connected fiber. The observation now follows from the fact that restriction induces a bijection from spin^{c}-structures to those over the 2-skeleton extending over the 3-skeleton. The same remark applies to bundles classified by BST OP or BSG if we define a complex structure to be a lift of the classifying map to BU.

(b) The Wu relations are known to hold for Poincaré complexes. In particular, for a compact, oriented 4-dimensional Poincaré complex X (without boundary) we have w_{2}(X) ∪ x = x ∪ x for all x ∈ H^{2}(X; ℤ_{2}). The usual argument [HH] then shows that W_{3}(X) = 0, so all such complexes admit spin^{c}-structures.

(c) As in the classical case, we have a canonical map

i : Bspin(H, w) → Bspin^{c}(H, w)

as the fiber of p_{SO(2)} (induced by inclusion of groups), inducing a map α : S(ξ, w) → S^{c}(ξ, w) that is equivariantly equivalent (when the domain is nonempty) to the Bockstein homomorphism β : H^{1}(X; ℤ_{2}) → H^{2}(X; ℤ). The image Im α is the set of spin^{c}-structures with c_{1} = 0, or equivalently, the set of conjugation-invariant structures. To verify that α has the stated equivariance and image, note that we can either consider i to be an inclusion into the fixed set of conjugation or replace it by a fibration p. Over each point in BH, i and p will restrict to the canonical inclusion and fibration ℝP^{∞} → ℂP^{∞}, respectively, both of which represent the unique nontrivial homotopy class of maps in [ℝP^{∞}, ℂP^{∞}]. For a fixed classifying map f̃ : ξ → ξ_{H}, spin structures s_{1}, s_{2} ∈ S(ξ, w) determine lifts f̂_{1}, f̂_{2} : X → Bspin(H, w). We can assume that these agree over the 0-skeleton and that p ∘ f̂_{1}, p ∘ f̂_{2} agree over the 1-skeleton, giving us obstruction cochains d(s_{1}, s_{2}) ∈ C^{1}(X; ℤ_{2}) and d(αs_{1}, αs_{2}) ∈ C^{2}(X; ℤ). Now d(αs_{1}, αs_{2}) evaluated on a 2-cell c is the element of π_{2}(ℂP^{∞}) ≅ ℤ given by p ∘ f̂_{2}(c) − p ∘ f̂_{1}(c). Since the boundary operator π_{2}(ℂP^{∞}) → π_{1}(ℝP^{1}) of p is multiplication by 2, the same coefficient is obtained as \frac{1}{2}⟨d(s_{1}, s_{2}), ∂c⟩ = ⟨βd(s_{1}, s_{2}), c⟩. Thus, we obtain the required equivariance d(αs_{1}, αs_{2}) = βd(s_{1}, s_{2}). To compute Im α, first note that any s ∈ Im α is conjugation-invariant (since i is) with c_{1} = 0. If S(ξ, w) is nonempty, fix s ∈ Im α and let s′ be any spin^{c}-structure that either is conjugationinvariant or satisfies c_{1}(s′) = 0. By Proposition 1, 2d(s, s′) = 0, so d(s, s′) ∈ Im β and s′ ∈ Im α. It now suffices to show that when S(ξ, w) is empty, no spin^{c}- structure has c_{1} = 0 or is conjugation-invariant. The first assertion is obvious since c_{1}|_{2} = w_{2} ≠ 0. For the remaining assertion, choose s ∈ S^{c}(ξ, w) with conjugate s̄. Since π_{1}(ℂP^{∞}, ℝP^{∞}) = 0, we can assume that the lift f̂: X → Bspin^{c}(H, w) determined by s maps the 1-skeleton X_{1} into i(Bspin(H, w)), which is fixed by conjugation. Thus, f̂ and its conjugate determine a difference cochain d(s, s̄) ∈ C^{2}(X, ℤ). Since π_{2}(ℂP^{∞}) → π_{2}(ℂP^{∞}, ℝP^{∞}) is multiplication by 2 on ℤ, we can change d(s, s̄) by any coboundary by changing f̂|X_{1} : X_{1} → i(Bspin(H, w)). Thus, if s = s̄ we can assume that d(s, s̄) = 0, so over each 2-cell, f̂ is conjugation-invariant up to homotopy rel ∂. But conjugation fixes only 0 in π_{2}(ℂP^{∞}, ℝP^{∞}), so f̂ can then be homotoped into i(Bspin(H, w)), i.e., s ∈ Im α.

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Department of Mathematics, University of Texas at Austin, Austin TX 78712

   E-mail address: gompf@math.utexas.edu