Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve

We consider the moduli space of rank two, odd degree, semi-stable Real vector bundles over a real curve, calculating the singular cohomology ring in odd and zero characteristic for most examples.

A Real C ∞ -vector bundle (E,τ ) over (Σ, τ ) is a complex C ∞ -vector bundle E → Σ equipped along with a smooth lift which is conjugate linear on fibres and such thatτ •τ = Id E . A Real holomorphic vector bundle is a Real C ∞ -vector bundle equipped with a holomorphic structure for whichτ is antiholomorphic. Quaternionic C ∞ /holomorphic vector bundles are defined similarly, except that one requiresτ •τ = − Id E , rather than Id E . Fix a complex C ∞ -vector bundle E → Σ with rank r and degree d such that gcd(r, d) = 1. The moduli space of semi-stable complex vector bundles is a compact, connected Kähler manifold (indeed a projective variety), that parametrizes isomorphism classes of semi-stable holomorphic vector bundles of C ∞ -type E.
Given a holomorphic vector bundle E over Σ, the conjugate pull-back τ * E is a holomorphic vector bundle over Σ of the same degree. This induces an anti-holomorphic, anti-symplectic involution on M (r, d) (which we also denote τ ) sending the equivalence class [E] to The fixed point set M (r, d) τ is a disjoint union of real, Lagrangian submanifolds. It was proven by Biswas-Huisman-Hurtubise [4] and independently by Schaffhauser [13,14], that the embeddings ιτ produce a diffeomorphism where the coproduct is indexed by C ∞ -isomorphism typesτ lifting τ . Hereafter, we abuse notation and identify M (r, d,τ ) = M (E,τ ) with its embedded image.
The Z 2 -Betti numbers of M (r, d,τ ) were calculated by Liu-Schaffhauser [10] and independently by the author [2] using the real Harder-Narasimhan stratification (described in Section 2.3). In the current paper, we use this stratification to compute the cohomology ring H * (M (2, d,τ ); k) for rank two bundles of odd degree d, and coefficient fields k of characteristic = 2, for most isomorphism typesτ (see Theorem 1.2) 1 .
The approach follows the general lines of approach laid out by Atiyah and Bott [1] in the complex case. We have the space Cτ = C(E,τ ) of Cauchy-Riemann operators on E that commute withτ . This is acted on by the group Gτ = G(E,τ ) of gauge transformations that commute withτ , and one is interested in the quotients under the action of this group. This leads us first to consider the Gτ -equivariant cohomology H * Gτ . For the total space of all operators, which is contractible, the equivariant cohomology is simply the singular cohomology of the classifying space BGτ ; one then must consider a stratification, and "remove" certain unstable strata, to obtain the equivariant cohomology of the semistable stratum Cτ ss ⊆ Cτ . A final step is to relate this equivariant cohomology to the ordinary cohomology of the moduli space M (E,τ ) := Cτ ss /Gτ . One issue is that the unstable strata no longer necessarily have orientable normal bundles, and so the cohomology of the Thom space of these strata, which is their contribution to the global cohomology, is no longer given by the Thom isomorphism. In the case we consider, that of rank two bundles, this turns out to be an advantage instead of a handicap, because it forces the Thom spaces to be acyclic. This yields the following result. Theorem 1.1. Let (E,τ ) be a Real/Quaternionic C ∞ -vector bundle over a real curve (Σ, τ ) of rank two. If the normal bundles of the unstable strata of the real Harder-Narasimhan stratification are all nonorientable, then there is a natural cohomology isomorphism If additionally E has odd degree, then H * (BG(E,τ ); k) ∼ = H * (M (E,τ ); k). 1 The condition that d is odd forces (Σ, τ ) to have real points and excludes the possibility that the bundle is quaternionic (see Remark 2.2), facts we exploit in the proof of our main result Theorem 1.2. However, many of our key results (first part of Theorem 1.1, Proposition 3.1 and Theorem 5.1) are independent of d and provide information about the moduli stack underlying M (2, d,τ ) when d is even. See for example Theorem 2.3, on the moduli stack of Quatenionic bundles.
We prove that the hypotheses of Theorem 1.1 hold except in certain special cases (see Proposition 3.1). This reduces our problem to calculating H * (BG(E,τ ); k), which is done using an Eilenberg-Moore spectral sequence in Section 5. For odd degree bundles, this determines the cohomology ring for most isomorphism types. Theorem 1.2. Let (Σ, τ ) be a real curve of genus g ≥ 2, let d be odd, and let k be a field of odd or zero characteristic. Then the cohomology ring H * (M (2, d,τ ); k) is an exterior algebra with g generators of degree 1 and (g − 1) of degree 3 for all but one exceptional C ∞ -typeτ which occurs only when (Σ, τ ) is a type I curve of even genus. If the genus g ≥ 4, then for that one exceptional type, H * (M (2, d,τ ); k) is not an exterior algebra.
The one exceptional C ∞ -type referred to in Theorem 1.2 is distinguished by the property that Eτ is non-orientable over every path component of Σ τ .
At first encounter, Theorem 1.2 is a little disappointing. For most C ∞ -types, the rational cohomology ring is simply an exterior algebra that does not depend on the real structure τ . In contrast, the Z 2 -cohomology has interesting Betti numbers that do depend on τ . On might be tempted to conclude that the odd and zero characteristic cohomology contains no interesting information.
However, we are led to a different conclusion if we consider moduli spaces of bundles with fixed determinant. Given a Real vector bundle (E,τ ), the determinant line bundle det(E) inherits a real structure fromτ . This gives rise to a natural fibre bundle map The fibre M Λ (r, d,τ ) := det −1 (Λ) is called the moduli space of Real vector bundles of fixed determinant Λ. The (finite) group, T r , of r-th roots of the trivial Real line bundle acts on M Λ (r, d,τ ) by tensor product. The following corollary follows from (and is equivalent to) Theorem 1.2. is an exterior algebra on (g − 1) generators of degree 3 for all but one exceptional C ∞ -typeτ which occurs only when (Σ, τ ) is a type I curve of even genus. If the genus g ≥ 4, then for the one exceptional type, H * (M Λ (2, d,τ ); k) is not an exterior algebra, as its first non-zero Betti number occurs in degree 2.
Calculations by the author (to appear elsewhere) show that H * (M Λ (2, d,τ ); Q) contains interesting information depending not only on τ , but also onτ . Corollary 1.3 implies that the interesting parts of H * (M Λ (2, d,τ ); Q) are generally not invariant under the action by T 2 . This is surprising, because for complex moduli spaces the analogous action is trivial: this was considered by Atiyah and Bott to be the main result of the famous Harder-Narasimhan paper [8] (see [1,Section 9]). We conclude that to understand the topology of M (r, d,τ ), it is important to study the fixed determinant moduli spaces M Λ (r, d,τ ). This is the focus of ongoing work by the author.

Topological classif ications
Let us begin by recalling the possibilities for a real structure on a Riemann surface. The nomenclature is that of [4]. The possible structures are: • Type 0 curves: On these, the real structure has no fixed points.
• Type I curves: For these, the real structure τ has a ≤ (g + 1) fixed circles such that the complement of the real points, Σ \ Σ τ , is disconnected. Necessarily, a ≡ g + 1 (mod 2).
• Type II curves: For these, the real structure τ has a ≤ g fixed circles such that Σ \ Σ τ is connected.
For all of these, one can write Σ as the union of two copies of a surface with boundary Σ 0 , with the identification taken along their boundaries, with τ interchanging the two copies. This is described in more detail in Section 5.1.
If (E,τ ) → (Σ, τ ) is a Real C ∞ -vector bundle of rank r, then the fixed point set Eτ forms an ordinary Real vector bundle over Σ τ , with fibre R r . Since Σ τ is a disjoint union of circles, the isomorphism type of Eτ is completely determined by the first Stiefel-Whitney class w 1 (Eτ ). In particular, for each path component S 1 ⊆ Σ τ we have It follows then that Eτ is nonorientable on an even number of path components, 1 if Eτ is nonorientable on an odd number of path components.
The classification of topological Real/Quaternionic vector bundles over a real curve (Σ, τ ) is as follows (see Propositions 4.1 and 4.2 of [4]).
Proposition 2.1. Topological Real vector bundles (E,τ ) over a real curve (Σ, τ ) are classified up to isomorphism by rank r, degree d and Stiefel-Whitney class w 1 (Eτ ) ∈ H 1 (Σ τ ; Z 2 ) subject to the condition that In particular, if the fixed point set Σ τ is a union of a ≥ 1 disjoint circles, then there are 2 a−1 isomorphism classes of Real C ∞ -vector bundles over (Σ, τ ) of any fixed rank and degree. Quaternionic vector bundles are classified by rank r and degree d, subject to the condition d ≡ r(g − 1) mod 2 and that Σ τ = ∅ if r is odd.
Remark 2.2. It follows from Proposition 2.1 that a rank two Quaternionic vector bundle must have even degree. This justifies our greater focus on Real bundles, since we are more interested in bundles with coprime rank and degree. It also follows that odd degree Real bundles can only occur over curves with real points, so that w 1 (Eτ ) may be non-zero. So for odd degree bundles, we need only consider curves of type I or II.

The complex Harder-Narasimhan stratif ication
A rank two holomorphic bundle over a curve E → Σ is called semi-stable if it does not contains any line sub-bundle of degree greater than d/2, where d is the degree of E. If E is not semi-stable, then we say it is unstable. Each unstable rank two bundle contains a unique line subbundle of maximum degree called the SCSS line sub-bundle (strongly contradicting semi-stability, see . Fix a C ∞ -vector bundle E of rank two and degree d over Σ. Let C = C(E) denote the space of Cauchy-Riemann operators on E. This is a contractible manifold modelled on a Sobolev completion of Ω 0,1 (Σ, End(E)). The Harder-Narasimhan stratification decomposes C into finite codimension submanifolds Here C ss is the subset of Cauchy-Riemann operators giving semi-stable holomorphic bundles; it is the open stratum. The set C d 1 is the subset of Cauchy-Riemann operators determining unstable bundles with SCSS line sub-bundle of degree d 1 ; it is a locally closed submanifold of complex codimension (2d 1 − d + g − 1) in C. 2 The complex gauge group G = G(E) acts naturally on C, preserving the stratification. The subgroup C * ≤ G acts trivially and the quotient G = G/C * acts effectively on C. Because C is contractible, the homotopy quotient C hG = EG × G C is a model for the classifying space BG.
The stratification (2.1) descends to a stratification The (topological) moduli stack of semistable, rank two, degree d bundles on Σ is the homotopy quotient If d is odd, then G acts freely on C ss and we may identify M(2, d) with the coarse moduli space which is a complex manifold of complex dimension 4g − 3 when g > 1 . Now suppose (Σ, τ ) is a real curve. Choose a real or quaternionic structureτ on the C ∞vector bundle E. LetĜ be the group of transformations of E generated by G andτ . Note thatĜ is independent of the choice ofτ , because for any other choiceτ , the compositioñ ττ ∈ G, so we have an equality of cosetsτ G =τ G. The natural action ofĜ on C preserves the stratification (2.1). This descends to a residual action of Z 2 =Ĝ/G on BG which preserves the stratification (2.2) and acts by anti-holomorphic involutions on the strata (we denote this involution by τ by abuse of notation). In particular, this means that the normal bundles of strata (2.2) are Real vector bundles with respect to τ .

The real Harder-Narasimhan stratif ication
Letτ denote a real or quaternionic structure on E and let Cτ = C(E,τ ) ⊂ C(E) denote the subspace of Cauchy-Riemann operators that are invariant underτ . It was explained in [2] that the Harder-Narasimhan stratification determines a stratification of Cτ where Cτ d 1 = C d 1 ∩ Cτ is a locally closed submanifold of real codimension (2d 1 − d + g − 1).
Define the real/quaternionic gauge group Gτ = G(E,τ ), to be the group of gauge transformations of E that commute withτ . The subgroup of scalars R * act trivially on Cτ and the quotient Gτ = Gτ /R * acts effectively on C(E,τ ) preserving the stratification (2.3). Since Cτ is contractible, we have a homotopy quotient BGτ = Cτ hGτ .

and (2.3) descends to a stratification
If d is odd, Gτ acts freely and we identify M(E,τ ) with the orbit space [14]). In some cases, the higher strata are actually empty.
is a rank two C ∞ -quaternionic bundle for which the fixed point set Σ τ is non-empty. Then the natural map M(E,τ ) → BGτ is a homotopy equivalence.
Proof . Suppose that E is an unstable rank two holomorphic bundle over Σ. If E were to admit a quaternionic structure lifting τ , then this would restrict to a quaternionic structure on the SCSS line sub-bundle (see [2, Section 2.2]). But this contradicts Proposition 2.1, because Σ τ is non-empty. It follows that every quaternionic lift of τ is semistable, so Cτ ss = Cτ and the result follows.
We are interested in determining the orientability of the normal bundle in the stratification (2.4). The inclusion Cτ → C respects the stratification, by construction, and is equivariant relative to the inclusion homomorphism Gτ → G, so it descends to a map i : BGτ → BG that respects the stratifications (2.2) and (2.4). The restriction of i to a map between corresponding strata determines a homotopy equivalence between (Cτ d 1 ) hGτ and the union of those path components of the fixed point set ((C d 1 ) hG ) τ corresponding toτ . This identifies the normal bundles to strata in (2.4) with the real points of the pull-backs of the normal bundles of (2.2) equipped with the real structure fromĜ/G = Z 2 . Thus we can determine orientability of normal bundles in (2.4) by studying the normal bundles of strata (2.2) with real structure τ .

Orientability of normal bundles
The goal of this section is to prove the following. 1. The degree of the bundle and the genus of the curve are of the same parity.
2. The type of the curve is I, and the Stiefel-Whitney class of the bundle vanishes on at least one component of the invariant curve Σ τ .
3. The type of the curve is II.
Conversely, if Σ τ = ∅ and none of the three conditions hold, then the normal bundles are all orientable.
To understand the normal bundles of our real strata (Cτ d 1 ) hGτ , we identify them as Z 2 -fixed point sets of the normal bundles for the complex strata (C d 1 ) hG . This approach considers all the C ∞ -types for the liftsτ of τ at once, and we must be careful to identify which path components corresponds to which liftsτ .
For unstable complex strata, we have a homotopy equivalence This can be explained as follows. Choose a decomposition into a sum of C ∞ -line bundles L 1 , L 2 of degrees d 1 , d 2 respectively. Let C(L i ) denote the space of Cauchy-Riemann operators on L i . The gauge group G(L i ) acts naturally on C(L i ) with orbit space This action is not free because the constant scalar transformations act trivially. Choose a base point p 0 ∈ Σ and denote G bas (L i ) ⊆ G(L i ) the subgroup of gauge transformations that act trivially on the fibre above p 0 . We have an internal direct product decomposition where the subgroup of scalar transformations C * acts trivially on C(L i ) and G bas (L i ) acts freely on C(L i ). The decomposition (3.2) induces morphisms C(L 1 ) × C(L 2 ) → C(E) and and determines a homotopy equivalence of homotopy quotients [1, Section 7] The subgroup G bas (L 1 )×G bas (L 2 ) acts freely and C * acts trivially, so the homotopy quotient (3.1) may be identified with the orbit space of C(L 1 ) × C(L 2 ) × EC * under the product action by the group G bas (L 1 ) × G bas (L 2 ) × C * . That is, The action of τ on (3.1) is a product action on each of the three factors.
is a union of path components, each homeomorphic to The different components correspond to the different C ∞ -types of the liftτ of the real structure τ to the bundle and of restrictions ofτ to the SCSS line sub-bundle. These C ∞ -types are classified by Stiefel-Whitney classes according to Section 2.1. If Σ τ has a ≥ 1 components, then the fixed point set (3.3) has 2 2a−2 components. The normal bundle N of (C d 1 ) hG is a complex vector bundle constructed in two stages as follows (for example, see the proof of Lemma 2 in [9]). Consider the vector bundle N over the Banach manifold C(L 1 ) × C(L 2 ) with fibres given by sheaf cohomology groups Since L * 1 ⊗ L 2 has negative degree, it admits no holomorphic sections for any choice of Cauchy-Riemann operator. By Riemann-Roch, it follows that N is a vector bundle of rank (2d 1 − d + g − 1). The action of G bas (L 1 ) × G(L 2 ) lifts naturally to N . The subgroup G bas (L 1 ) × G bas (L 2 ) acts freely, and the quotient yields a holomorphic vector bundle over Pic d 1 (Σ) × Pic d 2 (Σ). The subgroup C * acts with weight one on the fibres of N , hence also on N , giving rise to a vector bundle Recall that N has a real structure τ defined in Section 2.2.
Lemma 3.2. If g + d is even, then the normal bundle N τ is nonorientable on every path component of (C d 1 ) τ hG . Proof . Let x be a τ -fixed point in Pic d 1 (Σ) × Pic d 2 (Σ) and consider the restriction of N τ to {x} × BR * . Because C * acts by scalar multiplication of weight one on N , we have an isomorphism which is isomorphic to the Whitney sum of rank(N ) copies of the tautological bundle over BR * = RP ∞ . This is nonorientable if and only if rank(N ) is odd, which is true if and only if g + d is even.
The involution τ on N lifts to an involution τ of N by identifying N with the restriction of N to Pic The real structure on N induces one on det(N ). Since det(N ) is a holomorphic line bundle, this real structure is unique up to composition with a unit scalar (an analogue of Schur's lemma, see [4] or [12]), so the C ∞ -type of the lift is unique. Thus if we construct any real structure on det(N ), it must coincide with the one induced by τ up to C ∞ -isomorphism. Such a real structure has been carefully studied by Okonek-Teleman [12] (see also Cretois [5]).

Proof . Consider the bundle
This map is equivariant with respect to based gauge groups, and descends to φ as a map between orbit spaces. Thus the quotient bundle N is the pull-back of the quotient bundle V = V /G(L * 1 ⊗L 2 ) over Pic d 2 −d 1 (Σ). The determinant det(V ) is the determinant line bundle L considered by Okonek-Teleman.
As explained in [12, Section 1], L is isomorphic to the line bundle obtained by translating the geometric theta divisor Θ ⊂ Pic g−1 (Σ) by [(d 2 − d 1 − g + 1)p 0 ] ∈ Pic(Σ). If we choose a real base point p 0 ∈ Σ τ , then this divisor is sent to itself by τ , inducing a real structure on L. The Stiefel-Whitney class of L τ was calculated in Theorem 4.15 of [12] and the following lemma is a direct corollary. Lemma 3.4. Suppose that (Σ, τ ) is a real curve with real base point p 0 ∈ Σ τ , and suppose that g + d is odd.  Finally, we identify which path components correspond to which real structure on E.
Proof . If (E,τ ) decomposes as a sum of real bundles (L 1 ⊕ L 2 ,τ 1 ⊕τ 2 ), then Proof of Proposition 3.1. Sufficient condition 1 follows from Lemma 3.2. For sufficient conditions 2 and 3, consider the restriction of the map φ in Lemma 3.3 to path components of τ fixed point sets.
Choose a fixed element [D 1 ] ∈ M (1, d 1 ,τ 1 ). Then the map . It follows then that φ induces an injection on cohomology. In particular, the pullback of a nonorientable vector bundle by φ must be nonorientable. The result now follows from Lemmas 3.4 and 3.5.
For the converse statement, we have a type I curve equipped with a Real bundle (E,τ ) such that the genus and degree have opposite parity and w 1 (Eτ ) is non-vanishing on all path components of Σ τ . From Lemmas 3.4 and 3.5, we find that N τ is the pullback of an orientable bundle, hence is orientable. Finally, from the proof of Lemma 3.2, the quotient N τ = N τ /R * is also orientable.

Proof of Theorem 1.1
The (path components of) unstable strata of the real Harder-Narasimhan stratification (2.4) are homotopy equivalent to X := (S 1 ) 2g ×RP ∞ . The contribution of that stratum into Morse theory is through the relative cohomology groups H * (N, N 0 ) where N → X is an R n -vector bundle (the normal bundle) and N 0 ⊂ N is the complement of the zero section. If N is orientable then we have the Thom isomorphism but if N is not orientable we instead get the following. Proof . Suppose that N → X is a nonorientable vector bundle of rank n + 1. Using the long exact sequence of the pair, it is equivalent to show that the inclusion induced map is an isomorphism. The cohomology of the fibre H * (R n+1 \ 0) ∼ = H * (S n ) is k in degree 0 and n and is zero otherwise. Furthermore, the action of π 1 (X) on H 0 (S n ) is trivial and on H n (S n ) factors through a non-trivial homomorphism ρ : π 1 (X) → Z/2 because N is nonorientable. Thus if we denote by k ρ the locally constant k-sheaf twisted by ρ, then the Serre spectral sequence attached to the fibre bundle N 0 → X has E 2 -page satisfying E 0,q 2 ∼ = H q (X; k), E n,q 2 ∼ = H q (X; k ρ ) and E p,q 2 = 0 for p = 0, n. If we prove that H * (X; k ρ ) = 0, then the spectral sequence collapses and (4.1) is an isomorphism.
LetX → X denote the double cover of X defined by ρ : π 1 (X) → Z/2. Because we are working in a characteristic other than two, the transfer map defines an isomorphism where the direct sum decomposition is into the ±1-eigenspaces under the action by the deck transformation group Z/2. Since X = K(Z 2g × Z 2 , 1) is an Eilenberg-MacLane space,X = K(Γ, 1) for an index two subgroup Γ ⊂ Z 2g × Z 2 which by the classification of finitely generated abelian groups is isomorphic either to Z 2g × Z 2 or Z 2g . In either case, which combined with (4.2) implies that H * (X; k ρ ) = 0.
Proof of Theorem 1.1. Suppose that (E,τ ) is a rank two, real C ∞ -vector bundle over a real curve (Σ, τ ). Using the stratification (2.4), we construct the filtration By excision, for each i, where N i is the normal bundle of an unstable stratum. If the normal bundles of all positive codimension strata in the real Harder-Narasimhan stratification are nonorientable, then it follows from Proposition 4.1 that H * (Y i , Y i−1 ; k) = 0 and thus that inclusion induces an isomorphism Since this holds for all i ≥ 1, it follows by induction that If E has odd degree, the action of Gτ is free, so Finally, we relate equivariant cohomology with respect to Gτ and Gτ = Gτ /R * . Given any Gτ -space X on which the subgroup R * acts trivially, the natural map on Borel constructions EGτ × Gτ X → EGτ × Gτ X has homotopy fibre BR * = RP ∞ . Since H * (BR * ; k) is acyclic for coefficient fields k of characteristic not equal to 2, the Serre spectral sequence is trivial, yielding the isomorphism H * Gτ (X; k) = H * EGτ × Gτ X; k ∼ = H * EGτ × Gτ X; k = H * Gτ (X; k).
Applying this isomorphism when X = Cτ ss and when X is a point completes the proof.

Real gauge groups
In this section, we compute the cohomology ring of the classifying space BGτ . Recall that the Poincaré series of a space X is the generating function for its Betti numbers (X; k)). The goal of this section is to prove the following.
Theorem 5.1. Let (Σ, τ ) be a real curve of genus g with real points (i.e., Σ τ = ∅). Let Gτ = G(E,τ ) be the real gauge group of a rank 2 Real bundle (E,τ ) over (Σ, τ ), and let k be a coefficient field k of odd or zero characteristic. Then H * (BGτ ; k) is an exterior algebra with Poincaré series except in the following two special cases: 1. If the real structure on the curve is of type I and the restriction of Eτ to each component of Σ τ is nonorientable, then H * (BGτ ; k) is a free, graded commutative algebra with . Furthermore, if k has characteristic zero, then H * (BGτ ; k) is a free, graded commutative algebra.

Constructing the classifying space
Let (Σ, τ ) be a real curve of genus g and let (E,τ ) → (Σ, τ ) be a rank two Real C ∞ -vector bundle. In this subsection, we construct the classifying space BG(E,τ ) as a homotopy pullback. Instead of working with G(E,τ ) directly, we work with the subgroup of unitary gauge transformations U(E,τ ). The inclusion U(E,τ ) → G(E,τ ) is a homotopy equivalence, so they are interchangeable for our purposes. Up to homeomorphism, every real curve (Σ, τ ) can be constructed as follows. Let Σ 0 be a genusĝ surface with n boundary components, such that 2ĝ + n − 1 = g. Construct Σ by taking two copies of Σ 0 with opposite orientations, and gluing them together along their boundaries, attaching a ≤ n boundary circles to their counterpart using the identity map, and attaching the rest using the antipodal map. The involution τ simply transposes these two copies of Σ 0 . The resulting topological real curve has a fixed point circles. We get a type 0 curve if a = 0, a type I curve if a = n, and a type II curve if 0 < a < n.
Given a Real bundle (E,τ ) over (Σ, τ ), the unitary gauge symmetries that commute with τ (elements of U(E,τ )) are determined by their restriction to Σ 0 = Σ(ĝ, n). On the boundary circles, the gauge symmetries restrict to transformations of three types: (a) If τ | S 1 is the identity, and the restriction of Eτ to the circle is orientable, the gauge symmetries are maps from S 1 to O(2).
(b) If τ | S 1 is the identity, and the restriction of Eτ to the circle is nonorientable, the gauge symmetries are those of the Möbius R 2 -bundle.
(c) If τ | S 1 is a rotation by a half turn, our gauge symmetries satisfy g(θ) = g(θ + π) where the bar means entry-wise complex conjugation.
In each case, the restrictions of the gauge transformations to the boundaries are the so called real loop groups introduced in [2]. Let LUτ i 2 denote the real loop group over the ith circle of the boundary of Σ 0 = Σ(ĝ, n), with definitions varying from circle to circle, according to the behaviour ofτ .
• The LUτ i 2 are the real loop groups considered above, and ι is the product of inclusions LUτ i 2 → L 0 U 2 . Applying the classifying space functor yields a homotopy pull-back square BG(ĝ, n;τ 1 , . . . ,τ n ) Our strategy is to calculate H * (BG(ĝ, n;τ 1 , . . . ,τ n )) using the Eilenberg-Moore spectral sequence of diagram (5.2). We first need to understand the induced cohomology morphisms shown below Part of this was already calculated in [2].
Lemma 5.2. The map Bπ * fits into a commutative diagram Proof . This is proven in Lemma 4.4 of [2] for Z 2 -coefficients, but the proof actually works over Z and hence holds for any coefficient field.
Note in particular that K * , * is a free module over n i=1 Λ(b i ) ⊗ S(u i , w i ) and the cohomology of (K * , * , δ) is isomorphic to H * (BMaps 0 (Σ(ĝ, n), U 2 )). There is one real loop group of type (a) for each real component of Σ τ over which Eτ is trivial, one of type (b) for each real component for which Eτ is nonorientable, and a positive number of type (c) if and only if Σ \ Σ τ is connected.

Real loop groups
where deg(q) = 3 and deg(p) = 4. In the prior case, the restriction map sends b to q, w to p, and u to 0.
Proof . (a) We start with the case LUτ a 2 = LO 2 which is surely well known (the two-fold cover BLSO 2 → BLO 2 reduces the problem to the case of LSO 2 = LU 1 , which was considered in Lemma 4.4 of [2] for characteristic 2, but the proof works over any field). There is a canonical homotopy equivalence BLO 2 ∼ = Maps 0 (S 1 , BO 2 ) with the function space of maps homotopic to a constant map. Consider the evaluation map ev : S 1 × Maps 0 S 1 , BO 2 → BO 2 .
If p 1 ∈ H 4 (BO 2 ; k) denotes the first Pontryagin class, then we have an isomorphism H * (BLO 2 ) where denotes slant product with respect to the homology classes [pt], [S 1 ].
The two remaining cases can be realized up to isomorphism as twisted loop groups (see Baird [3]). Let I = [0, 1] be the unit interval, let G be a compact Lie group, and let σ ∈ Aut(G) be an automorphism. Then the associated twisted loop group is where σ is an orientation reversing orthogonal change of basis.
Note that σ restricts to an automorphism of SO 2 . It was proven in [3, Proposition 7.6] that for coefficient fields of characteristic other than two, (c) Consider the map I → S 1 that embeds I as a half circle. Then restriction determines an isomorphism where σ is entry-wise complex conjugation. By [3, Corollary 7.5], the inclusions LO 2 ⊂ L σ U 2 induces an isomorphism Because (5.5) is induced by including I as a half circle in S 1 , we have a commutative diagram where the vertical arrows are induced by inclusion and f * is induced by a 2-fold covering map f : S 1 → S 1 . Functoriality properties of the slant product imply that ev * (p 1 )) = 2q.
Since 2 is invertible, we can simply relabel 2q as q as an element of H * (BLUτ c 2 ) completing the proof.

The spectral sequence
We refer the reader to [2,Appendix A] or McLeary [11,Section 7.1] for background on the Eilenberg-Moore spectral sequence.
If b > 0, then the bigraded algebra EM * , * 2 is generated by homogenous elements lying in EM 0,q 2 or EM −1,q 2 for some q (i.e., the 0th and −1th columns). For degree reasons the generators must survive until infinity, so the spectral sequence must collapse. Since EM * , * 2 is a free gradedcommutative algebra and an associated graded algebra of H * (BGτ ), we deduce that H * (BGτ ) is a free super commutative algebra isomorphic to EM * , * 2 . If b = 0, then the preceding argument still works for coefficient fields of characteristic zero fields, because in that case Γ(z) = S(z). The universal coefficient theorem then implies that the spectral sequence collapses for odd characteristic fields as well.