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.
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irk-123456789-1477272019-02-16T01:26:35Z Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve Baird, T.J. 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. 2016 Article Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve / T.J. Baird // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D30; 55R10; 55T20 DOI:10.3842/SIGMA.2016.072 http://dspace.nbuv.gov.ua/handle/123456789/147727 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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. |
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Baird, T.J. Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve Symmetry, Integrability and Geometry: Methods and Applications |
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Baird, T.J. |
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Baird, T.J. |
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Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve |
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Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve |
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Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve |
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Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve |
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Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve |
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cohomology of the moduli space of rank two, odd degree vector bundles over a real curve |
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Інститут математики НАН України |
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Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve / T.J. Baird // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 14 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT bairdtj cohomologyofthemodulispaceofranktwoodddegreevectorbundlesoverarealcurve |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 072, 18 pages
Cohomology of the Moduli Space of Rank Two,
Odd Degree Vector Bundles over a Real Curve
Thomas John BAIRD
Department of Mathematics and Statistics, Memorial University of Newfoundland,
St. John’s, NL, A1C 5S7, Canada
E-mail: tbaird@mun.ca
URL: https://thomasjohnbaird.com
Received October 30, 2015, in final form July 20, 2016; Published online July 22, 2016
http://dx.doi.org/10.3842/SIGMA.2016.072
Abstract. We consider the moduli space of rank two, odd degree, semi-stable Real vec-
tor bundles over a real curve, calculating the singular cohomology ring in odd and zero
characteristic for most examples.
Key words: moduli space of vector bundles; gauge groups; real curves
2010 Mathematics Subject Classification: 53D30; 55R10; 55T20
1 Introduction
A real curve (Σ, τ) is a closed Riemann surface Σ of genus g, endowed with an anti-holomorphic
involution τ . The real points of (Σ, τ) are those fixed by τ ,
Στ = {p ∈ Σ | τ(p) = p}.
A Real C∞-vector bundle (E, τ̃) over (Σ, τ) is a complex C∞-vector bundle E → Σ equipped
along with a smooth lift
E
τ̃ //
��
E
��
Σ
τ // Σ,
which is conjugate linear on fibres and such that τ̃ ◦ τ̃ = IdE . A Real holomorphic vector
bundle is a Real C∞-vector bundle equipped with a holomorphic structure for which τ̃ is anti-
holomorphic. Quaternionic C∞/holomorphic vector bundles are defined similarly, except that
one requires τ̃ ◦ τ̃ = − IdE , rather than IdE .
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
M(E) = M(r, d)
is a compact, connected Kähler manifold (indeed a projective variety), that parametrizes iso-
morphism classes of semi-stable holomorphic vector bundles of C∞-type E.
Similarly, fix a Real/Quaternionic C∞-vector bundle (E, τ̃) with coprime rank and degree.
The moduli space of semi-stable Real/Quaternionic vector bundles
M(E, τ̃) = M(r, d, τ̃)
mailto:tbaird@mun.ca
https://thomasjohnbaird.com
http://dx.doi.org/10.3842/SIGMA.2016.072
2 T.J. Baird
is a compact, connected manifold that parametrizes isomorphism classes of semi-stable
Real/Quaternionic holomorphic vector bundles with C∞-isomorphism type (E, τ̃). The forgetful
map
ιτ̃ : M(r, d, τ̃) ↪→M(r, d),
embeds M(r, d, τ̃) as a real, Lagrangian submanifold of M(r, d).
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
τ([E ]) = [τ∗E ].
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 embed-
dings ιτ̃ produce a diffeomorphism∐
τ̃
M(r, d, τ̃) ∼= M(r, d)τ ,
where the coproduct is indexed by C∞-isomorphism types τ̃ lifting τ . Hereafter, we abuse
notation and identify M(r, d, τ̃) = M(E, τ̃) with its embedded image.
The Z2-Betti numbers of M(r, d, τ̃) were calculated by Liu–Schaffhauser [10] and indepen-
dently by the author [2] using the real Harder–Narasimhan stratification (described in Sec-
tion 2.3). In the current paper, we use this stratification to compute the cohomology ring
H∗(M(2, d, τ̃); k) for rank two bundles of odd degree d, and coefficient fields k of characteristic
6= 2, for most isomorphism types τ̃ (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 C τ̃ = C(E, τ̃) of Cauchy–Riemann operators on E that commute
with τ̃ . This is acted on by the group G τ̃ = G(E, τ̃) of gauge transformations that commute
with τ̃ , and one is interested in the quotients under the action of this group. This leads us first
to consider the G τ̃ -equivariant cohomology H∗Gτ̃ . For the total space of all operators, which is
contractible, the equivariant cohomology is simply the singular cohomology of the classifying
space BG τ̃ ; one then must consider a stratification, and “remove” certain unstable strata, to
obtain the equivariant cohomology of the semistable stratum C τ̃ss ⊆ C τ̃ . A final step is to relate
this equivariant cohomology to the ordinary cohomology of the moduli space M(E, τ̃) := C τ̃ss/G τ̃ .
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, τ̃) 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 stratifica-
tion are all nonorientable, then there is a natural cohomology isomorphism
H∗(BG(E, τ̃); k) ∼= H∗G(E,τ̃)(Css(E, τ̃); k)
for coefficient fields k of characteristic 6= 2. If additionally E has odd degree, then
H∗(BG(E, τ̃); k) ∼= H∗(M(E, τ̃); k).
1The 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, τ̃) when d is even. See for example Theorem 2.3, on the
moduli stack of Quatenionic bundles.
Cohomology of the Moduli Space of Rank Two 3
We prove that the hypotheses of Theorem 1.1 hold except in certain special cases (see Propo-
sition 3.1). This reduces our problem to calculating H∗(BG(E, τ̃); 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, τ̃); k) is an exterior algebra
with g generators of degree 1 and (g−1) of degree 3 for all but one exceptional C∞-type τ̃ 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, τ̃); k) is not an exterior algebra.
The one exceptional C∞-type referred to in Theorem 1.2 is distinguished by the property
that E τ̃ is non-orientable over every path component of Στ .
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 Z2-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, τ̃), the determinant line bundle det(E)
inherits a real structure from τ̃ . This gives rise to a natural fibre bundle map
det : M(r, d, τ̃)→M(1, d, τ̃).
The fibre MΛ(r, d, τ̃) := det−1(Λ) is called the moduli space of Real vector bundles of fixed
determinant Λ. The (finite) group, Tr, of r-th roots of the trivial Real line bundle acts on
MΛ(r, d, τ̃) by tensor product. The following corollary follows from (and is equivalent to) The-
orem 1.2.
Corollary 1.3. Under the hypotheses of Theorem 1.2, the ring of T2-invariants
H∗(MΛ(2, d, τ̃); k)T2
is an exterior algebra on (g − 1) generators of degree 3 for all but one exceptional C∞-type τ̃
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∗(MΛ(2, d, τ̃); 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∗(MΛ(2, d, τ̃);Q) contains in-
teresting information depending not only on τ , but also on τ̃ . Corollary 1.3 implies that the
interesting parts of H∗(MΛ(2, d, τ̃);Q) are generally not invariant under the action by T2. This
is surprising, because for complex moduli spaces the analogous action is trivial: this was con-
sidered 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, τ̃), it is important
to study the fixed determinant moduli spaces MΛ(r, d, τ̃). This is the focus of ongoing work by
the author.
2 Preliminaries
2.1 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:
4 T.J. Baird
• 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 \ Στ , 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 \ Στ is
connected.
For all of these, one can write Σ as the union of two copies of a surface with boundary Σ0, 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, τ̃)→ (Σ, τ) is a Real C∞-vector bundle of rank r, then the fixed point set E τ̃ forms an
ordinary Real vector bundle over Στ , with fibre Rr. Since Στ is a disjoint union of circles, the
isomorphism type of E τ̃ is completely determined by the first Stiefel–Whitney class w1(E τ̃ ). In
particular, for each path component S1 ⊆ Στ we have
w1
(
E τ̃ |S1
)(
S1
)
=
{
0 if E τ̃ |S1 is orientable (hence trivial),
1 if E τ̃ |S1 is nonorientable (hence a Möbius bundle).
It follows then that
w1
(
E τ̃
)(
Στ
)
=
{
0 if E τ̃ is nonorientable on an even number of path components,
1 if E τ̃ 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, τ̃) over a real curve (Σ, τ) are classified
up to isomorphism by rank r, degree d and Stiefel–Whitney class w1(E τ̃ ) ∈ H1(Στ ;Z2) subject
to the condition that
d ≡ w1
(
E τ̃
)(
Στ
)
mod 2.
In particular, if the fixed point set Στ is a union of a ≥ 1 disjoint circles, then there are 2a−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 Στ = ∅ 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 w1(E τ̃ ) may be non-zero. So for odd degree bundles,
we need only consider curves of type I or II.
2.2 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
Harder–Narasimhan [8]).
Cohomology of the Moduli Space of Rank Two 5
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 Ω0,1(Σ,End(E)). The Harder–Narasimhan stratification decomposes C into finite
codimension submanifolds
C = Css ∪
( ⋃
d1>d/2
Cd1
)
. (2.1)
Here Css is the subset of Cauchy–Riemann operators giving semi-stable holomorphic bundles;
it is the open stratum. The set Cd1 is the subset of Cauchy–Riemann operators determining
unstable bundles with SCSS line sub-bundle of degree d1; it is a locally closed submanifold
of complex codimension (2d1 − 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
ChG = EG ×G C is a model for the classifying space BG.
BG = ChG .
The stratification (2.1) descends to a stratification
BG = (Css)hG ∪
( ⋃
d1>d/2
(Cd1)hG
)
. (2.2)
The (topological) moduli stack of semistable, rank two, degree d bundles on Σ is the homotopy
quotient
M(2, d) =M(E) = (Css)hG .
If d is odd, then G acts freely on Css and we may identifyM(2, d) with the coarse moduli space
M(2, d) = Css/G,
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 structure τ̃ on the C∞-
vector bundle E. Let Ĝ be the group of transformations of E generated by G and τ̃ . Note
that Ĝ is independent of the choice of τ̃ , because for any other choice τ̃ ′, the composition
τ̃ τ̃ ′ ∈ G, so we have an equality of cosets τ̃G = τ̃ ′G. The natural action of Ĝ on C preserves
the stratification (2.1). This descends to a residual action of Z2 = 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 .
2.3 The real Harder–Narasimhan stratification
Let τ̃ denote a real or quaternionic structure on E and let C τ̃ = C(E, τ̃) ⊂ C(E) denote the
subspace of Cauchy–Riemann operators that are invariant under τ̃ . It was explained in [2] that
the Harder–Narasimhan stratification determines a stratification of C τ̃
C τ̃ = C τ̃ss ∪
( ⋃
d1>d/2
C τ̃d1
)
. (2.3)
where C τ̃d1 = Cd1 ∩ C τ̃ is a locally closed submanifold of real codimension (2d1 − d+ g − 1).
2By work of Atiyah–Bott [1] and Daskalopoulos [6], (2.1) is also the Morse stratification induced by the
Yang–Mills functional, but we won’t use this fact.
6 T.J. Baird
Define the real/quaternionic gauge group G τ̃ = G(E, τ̃), to be the group of gauge transfor-
mations of E that commute with τ̃ . The subgroup of scalars R∗ act trivially on C τ̃ and the
quotient G τ̃ = G τ̃/R∗ acts effectively on C(E, τ̃) preserving the stratification (2.3). Since C τ̃ is
contractible, we have a homotopy quotient
BG τ̃ = C τ̃
hGτ̃ .
and (2.3) descends to a stratification
BG τ̃ = (C τ̃ss)hGτ̃ ∪
( ⋃
d1>d/2
(C τ̃d1)
hGτ̃
)
. (2.4)
The (topological) moduli stack M(E, τ̃) is the homotopy quotient
M(E, τ̃) =
(
C τ̃ss
)
hGτ̃
.
If d is odd, G τ̃ acts freely and we identify M(E, τ̃) with the orbit space
M(E, τ̃) = C τ̃ss/G τ̃
which is a compact manifold of real dimension 4g − 3 when g > 1 that embeds in M(2, d) as
a path component of the fixed point set M(2, d)τ (see [14]).
In some cases, the higher strata are actually empty.
Theorem 2.3. Suppose that (E, τ̃) → (Σ, τ) is a rank two C∞-quaternionic bundle for which
the fixed point set Στ is non-empty. Then the natural map M(E, τ̃) ↪→ BG τ̃ 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 Στ is
non-empty. It follows that every quaternionic lift of τ is semistable, so C τ̃ss = C τ̃ and the result
follows. �
We are interested in determining the orientability of the normal bundle in the stratifica-
tion (2.4). The inclusion C τ̃ ↪→ C respects the stratification, by construction, and is equivariant
relative to the inclusion homomorphism G τ̃ ↪→ G, so it descends to a map
i : BG τ̃ → BG
that respects the stratifications (2.2) and (2.4). The restriction of i to a map between corre-
sponding strata
(C τ̃d1)
hGτ̃ → (Cd1)hG
determines a homotopy equivalence between (C τ̃d1)
hGτ̃ and the union of those path components
of the fixed point set ((Cd1)hG)τ corresponding to τ̃ . 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 from Ĝ/G = Z2. Thus we can determine orientability of normal bundles in (2.4)
by studying the normal bundles of strata (2.2) with real structure τ .
Cohomology of the Moduli Space of Rank Two 7
3 Orientability of normal bundles
The goal of this section is to prove the following.
Proposition 3.1. The normal bundles of the stratification (2.4) are all nonorientable if any of
the following holds:
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 Στ .
3. The type of the curve is II.
Conversely, if Στ 6= ∅ and none of the three conditions hold, then the normal bundles are all
orientable.
To understand the normal bundles of our real strata (C τ̃d1)
hGτ̃ , we identify them as Z2-fixed
point sets of the normal bundles for the complex strata (Cd1)hG . This approach considers all the
C∞-types for the lifts τ̃ of τ at once, and we must be careful to identify which path components
corresponds to which lifts τ̃ .
For unstable complex strata, we have a homotopy equivalence
(Cd1)hG = Picd1(Σ)× Picd2(Σ)× CP∞, (3.1)
where d2 = d− d1. This can be explained as follows. Choose a decomposition
E = L1 ⊕ L2 (3.2)
into a sum of C∞-line bundles L1, L2 of degrees d1, d2 respectively. Let C(Li) denote the space
of Cauchy–Riemann operators on Li. The gauge group G(Li) acts naturally on C(Li) with orbit
space
C(Li)/G(Li) = Picdi(Σ).
This action is not free because the constant scalar transformations act trivially. Choose a base
point p0 ∈ Σ and denote Gbas(Li) ⊆ G(Li) the subgroup of gauge transformations that act
trivially on the fibre above p0. We have an internal direct product decomposition
G(Li) = Gbas(Li)× C∗,
where the subgroup of scalar transformations C∗ acts trivially on C(Li) and Gbas(Li) acts freely
on C(Li). The decomposition (3.2) induces morphisms C(L1)× C(L2) ↪→ C(E) and
Gbas(L1)× G(L2) ∼= Gbas(L1)× Gbas(L2)× C∗ ↪→ G,
and determines a homotopy equivalence of homotopy quotients [1, Section 7]
(C(L1)× C(L2))h(Gbas(L1)×Gbas(L2)×C∗) ∼= (Cd1)hG .
The subgroup Gbas(L1)×Gbas(L2) acts freely and C∗ acts trivially, so the homotopy quotient (3.1)
may be identified with the orbit space of C(L1)×C(L2)×EC∗ under the product action by the
group Gbas(L1)× Gbas(L2)× C∗. That is,
(Cd1)hG
∼= C(L1)/Gbas(L1)× C(L2)/Gbas(L2)×BC∗
∼= Picd1(Σ)× Picd2(Σ)× CP∞.
8 T.J. Baird
The action of τ on (3.1) is a product action on each of the three factors. In terms of divisor
classes [Di] ∈ Picdi(Σ) and a projective point [v] ∈ CP∞ the action sends ([D1], [D2], [v]) to
([τ(D1)], [τ(D2)], [v]) (this involution on the Picdi(Σ) factors was studied by Gross–Harris [7]).
The fixed point set, denoted(
Picd1(Σ)× Picd−d1(Σ)× CP∞
)Z2 = Picd1(Σ)τ × Picd−d1(Σ)τ × RP∞ (3.3)
is a union of path components, each homeomorphic to(
S1
)g × (S1
)g × RP∞.
The different components correspond to the different C∞-types of the lift τ̃ of the real struc-
ture τ to the bundle and of restrictions of τ̃ to the SCSS line sub-bundle. These C∞-types are
classified by Stiefel–Whitney classes according to Section 2.1. If Στ has a ≥ 1 components, then
the fixed point set (3.3) has 22a−2 components.
The normal bundle N of (Cd1)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(L1)× C(L2) with fibres given by sheaf cohomology groups
N ′′(∂̄1,∂̄2) = H1
(
L∗1 ⊗ L2, ∂̄
∗
1 ⊗ ∂̄2
)
.
Since L∗1⊗L2 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 (2d1 − d+
g− 1). The action of Gbas(L1)×G(L2) lifts naturally to N ′′. The subgroup Gbas(L1)×Gbas(L2)
acts freely, and the quotient yields a holomorphic vector bundle
N ′ = N ′′/Gbas(L1)× Gbas(L2)
over Picd1(Σ)×Picd2(Σ). The subgroup C∗ acts with weight one on the fibres of N ′′, hence also
on N ′, giving rise to a vector bundle
N = N ′hC∗ = N ′ ×C∗ EC∗
over Picd1(Σ)× Picd2(Σ)× CP∞.
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 (Cd1)τ
hG.
Proof. Let x be a τ -fixed point in Picd1(Σ) × Picd2(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
N τ |{x}×BR∗ = N ′τ
′
x ×R∗ ER∗,
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 Picd1(Σ)×Picd2(Σ)×{[v]} for some fixed point {[v]} ∈ RP∞ ⊂ CP∞. If the degree d and
the genus g have different parity, then N τ is nonorientable if and only if N ′τ is nonorientable,
or equivalently the first Stiefel–Whitney class w1(N ′τ ) does not vanish. The equalities
w1(N ′τ ) = w1(det(N ′τ )) = w1(det(N ′)τ )
permits us to work with the determinant line bundle det(N ′).
Cohomology of the Moduli Space of Rank Two 9
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]).
Lemma 3.3. Let
φ : Picd1(Σ)× Picd2(Σ)→ Picd2−d1(Σ)
be the map on divisor classes φ([D1], [D2]) = [D2 − D1]. Then there is an isomorphism of
holomorphic line bundles
det(N ′) ∼= φ∗(L),
where L := det ind δLp0 is the determinant line bundle considered by Okonek–Teleman [12, Sec-
tion 1].
Proof. Consider the bundle V ′′ over C(L∗1⊗L2) with fibres V ′′
∂̄
= H1(L∗1⊗L2, ∂̄). Clearly N ′′ is
the pull-back of V ′′ under the map C(L1)×C(L2)→ C(L∗1⊗L2) that sends (∂̄1, ∂̄2) to ∂̄∗1⊗∂̄2. 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⊗L2)
over Picd2−d1(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 ⊂ Picg−1(Σ) by [(d2− d1− g+ 1)p0] ∈ Pic(Σ). If we choose a real
base point p0 ∈ Στ , then this divisor is sent to itself by τ , inducing a real structure on L. The
Stiefel–Whitney class of Lτ 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 p0 ∈ Στ , and suppose that
g + d is odd.
(i) If (Σ, τ) is type I, then w1(Lτ ) vanishes only on the path component M(L, τ̃L)⊆Picd2−d1(Σ)τ
for which the restrictions Lτ̃L |S1 are nonorientable for all real circles S1 ⊆ Στ̃ .
(ii) If (Σ, τ) is type II, then w1(Lτ ) does not vanish on any path component of Picd2−d1(Σ)τ .
Finally, we identify which path components correspond to which real structure on E.
Lemma 3.5. Let (E, τ̃) → (Σ, τ) be a rank two C∞-real bundle. The map φ sends all path
components of Picd1(Σ)τ × Picd2(Σ)τ corresponding to an unstable stratum of C τ̃ to a unique
path component of M(L, τ̃L) ⊆ Picd2−d1(Σ)τ for which the Stiefel–Whitney classes agree:
w1
(
Lτ̃L
)
= w1
(
E τ̃
)
in H1(Στ ;Z/2).
Proof. If (E, τ̃) decomposes as a sum of real bundles (L1 ⊕ L2, τ̃1 ⊕ τ̃2), then
w1
(
E τ̃
)
= w1
(
(L1 ⊗ L2)(τ̃1⊗τ̃2)
)
= w1
(
(L∗1 ⊗ L2)(τ̃∗1⊗τ̃2)
)
. �
10 T.J. Baird
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.
φ′ : M(1, d1, τ̃1)×M(1, d2, τ̃2)→M(1, d2 − d1, τ̃
∗
1 ⊗ τ̃2).
Choose a fixed element [D1] ∈M(1, d1, τ̃1). Then the map
ψ : M(1, d2 − d1, τ̃
∗
1 ⊗ τ̃2)→M(1, d1, τ̃1)×M(1, d2, τ̃2)
sending ψ([D]) = ([D1], [D + D1]) is a left inverse of φ′ (i.e., φ′ ◦ ψ is the identity map on
M(1, d2− d1, τ̃
∗
1 ⊗ τ̃2). 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, τ̃)
such that the genus and degree have opposite parity and w1(E τ̃ ) is non-vanishing on all path
components of Στ . 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. �
4 Proof of Theorem 1.1
The (path components of) unstable strata of the real Harder–Narasimhan stratification (2.4) are
homotopy equivalent to X := (S1)2g×RP∞. The contribution of that stratum into Morse theory
is through the relative cohomology groups H∗(N,N0) where N → X is an Rn-vector bundle (the
normal bundle) and N0 ⊂ N is the complement of the zero section. If N is orientable then we
have the Thom isomorphism
H∗(N,N0) ∼= H∗−n(X),
but if N is not orientable we instead get the following.
Proposition 4.1. Let X := (S1)2g × RP∞ and k a field of characteristic 6= 2. If N → X is
a nonorientable real vector bundle, then H∗(N,N0; k) = 0.
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
H∗(X) = H∗(N)→ H∗(N0) (4.1)
is an isomorphism.
The cohomology of the fibre H∗(Rn+1 \ 0) ∼= H∗(Sn) is k in degree 0 and n and is zero
otherwise. Furthermore, the action of π1(X) on H0(Sn) is trivial and on Hn(Sn) factors through
a non-trivial homomorphism ρ : π1(X) → Z/2 because N is nonorientable. Thus if we denote
by kρ the locally constant k-sheaf twisted by ρ, then the Serre spectral sequence attached to the
fibre bundle N0 → X has E2-page satisfying E0,q
2
∼= Hq(X; k), En,q2
∼= Hq(X; kρ) and Ep,q2 = 0
for p 6= 0, n. If we prove that H∗(X; kρ) = 0, then the spectral sequence collapses and (4.1) is
an isomorphism.
Let X̃ → X denote the double cover of X defined by ρ : π1(X) → Z/2. Because we are
working in a characteristic other than two, the transfer map defines an isomorphism
H∗(X̃; k) ∼= H∗(X; k)⊕H∗(X; kρ), (4.2)
Cohomology of the Moduli Space of Rank Two 11
where the direct sum decomposition is into the ±1-eigenspaces under the action by the deck
transformation group Z/2.
Since X = K(Z2g×Z2, 1) is an Eilenberg–MacLane space, X̃ = K(Γ, 1) for an index two sub-
group Γ ⊂ Z2g×Z2 which by the classification of finitely generated abelian groups is isomorphic
either to Z2g × Z2 or Z2g. In either case,
H∗(X̃; k) ∼= H∗(K(Γ, 1); k) ∼= H∗(X; k),
which combined with (4.2) implies that H∗(X; kρ) = 0. �
Proof of Theorem 1.1. Suppose that (E, τ̃) is a rank two, real C∞-vector bundle over a real
curve (Σ, τ). Using the stratification (2.4), we construct the filtration(
C τ̃ss
)
hGτ̃ = Y0 ⊆ Y1 ⊆ · · · ⊆ Y∞ = BG τ̃ ,
where
Yi =
(
C τ̃ss
)
hGτ̃ ∪
( ⋃
d/2+i≥d1
(C τ̃d1)
hGτ̃
)
.
By excision, for each i,
H∗(Yi, Yi−1; k) ∼= H∗(Ni, (Ni)0; k),
where Ni is the normal bundle of an unstable stratum. If the normal bundles of all positive codi-
mension strata in the real Harder–Narasimhan stratification are nonorientable, then it follows
from Proposition 4.1 that H∗(Yi, Yi−1; k) = 0 and thus that inclusion induces an isomorphism
H∗(Yi−1; k) ∼= H∗(Yi; k).
Since this holds for all i ≥ 1, it follows by induction that
H∗Gτ̃
(
C τ̃ss; k
)
= H∗(Y0; k) ∼= H∗(Y∞; k) = H∗
(
BG τ̃ ; k
)
.
If E has odd degree, the action of G τ̃ is free, so
H∗Gτ̃
(
C τ̃ss; k
) ∼= H∗
(
C τ̃ss/G τ̃ ; k
)
= H∗(M(E, τ̃); k).
Finally, we relate equivariant cohomology with respect to G τ̃ and G τ̃ = G τ̃/R∗. Given any
G τ̃ -space X on which the subgroup R∗ acts trivially, the natural map on Borel constructions
EG τ̃ ×Gτ̃ X → EG τ̃ ×Gτ̃ X
has homotopy fibre BR∗ = RP∞. Since H∗(BR∗; k) is acyclic for coefficient fields k of charac-
teristic not equal to 2, the Serre spectral sequence is trivial, yielding the isomorphism
H∗Gτ̃ (X; k) = H∗
(
EG τ̃ ×Gτ̃ X; k
) ∼= H∗
(
EG τ̃ ×Gτ̃ X; k
)
= H∗Gτ̃ (X; k).
Applying this isomorphism when X = C τ̃ss and when X is a point completes the proof. �
12 T.J. Baird
5 Real gauge groups
In this section, we compute the cohomology ring of the classifying space BG τ̃ . Recall that the
Poincaré series of a space X is the generating function for its Betti numbers
Pt(X) =
∞∑
i=0
bit
i,
where bi = dim(H i(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., Στ 6= ∅). Let
G τ̃ = G(E, τ̃) be the real gauge group of a rank 2 Real bundle (E, τ̃) over (Σ, τ), and let k be
a coefficient field k of odd or zero characteristic. Then H∗(BG τ̃ ; k) is an exterior algebra with
Poincaré series
Pt(BG τ̃ ) = (1 + t)g
(
1 + t3
)g−1
,
except in the following two special cases:
1. If the real structure on the curve is of type I and the restriction of E τ̃ to each compo-
nent of Στ is nonorientable, then H∗(BG τ̃ ; k) is a free, graded commutative algebra with
Pt(BG τ̃ ) = (1 + t)g(1 + t3)g/(1− t2).
2. If the restriction of E τ̃ to each component of Στ is orientable, then Pt(BG τ̃ ) = (1+ t)g(1+
t3)g/(1− t4). Furthermore, if k has characteristic zero, then H∗(BG τ̃ ; k) is a free, graded
commutative algebra.
5.1 Constructing the classifying space
Let (Σ, τ) be a real curve of genus g and let (E, τ̃) → (Σ, τ) be a rank two Real C∞-vector
bundle. In this subsection, we construct the classifying space BG(E, τ̃) as a homotopy pull-
back. Instead of working with G(E, τ̃) directly, we work with the subgroup of unitary gauge
transformations U(E, τ̃). The inclusion U(E, τ̃) ↪→ G(E, τ̃) 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 Σ0 be
a genus ĝ surface with n boundary components, such that 2ĝ+n−1 = g. Construct Σ by taking
two copies of Σ0 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 Σ0. 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, τ̃) over (Σ, τ), the unitary gauge symmetries that commute with
τ̃ (elements of U(E, τ̃)) are determined by their restriction to Σ0 = Σ(ĝ, n). On the boundary
circles, the gauge symmetries restrict to transformations of three types:
(a) If τ |S1 is the identity, and the restriction of E τ̃ to the circle is orientable, the gauge
symmetries are maps from S1 to O(2).
(b) If τ |S1 is the identity, and the restriction of E τ̃ to the circle is nonorientable, the gauge
symmetries are those of the Möbius R2-bundle.
(c) If τ |S1 is a rotation by a half turn, our gauge symmetries satisfy g(θ) = g(θ + π) where
the bar means entry-wise complex conjugation.
Cohomology of the Moduli Space of Rank Two 13
In each case, the restrictions of the gauge transformations to the boundaries are the so called
real loop groups introduced in [2]. Let LU τ̃i2 denote the real loop group over the ith circle of
the boundary of Σ0 = Σ(ĝ, n), with definitions varying from circle to circle, according to the
behaviour of τ̃ .
As explained in [2, Section 6.1], the unitary real gauge group U(E, τ̃) can be constructed up
to isomorphism as the pull-back G(ĝ, n; τ̃1, . . . , τ̃n) of topological groups
G(ĝ, n; τ̃1, . . . , τ̃n) //
��
Maps0(Σ(ĝ, n), U2)
π
��
n∏
i=1
LU τ̃i2
ι //
n∏
i=1
L0U2.
(5.1)
In this diagram
• Maps0(Σ(ĝ, n), U2) is the group of maps from Σ(ĝ, n) to U2 that send each boundary circle
of Σ(ĝ, n) to a contractible loop in U2.
• L0U2 is the group of contractible loops f : S1 → U2 and π is defined by restriction from
Σ(ĝ, n) to ∂Σ(ĝ, n).
• The LU τ̃i2 are the real loop groups considered above, and ι is the product of inclusions
LU τ̃i2 ↪→ L0U2.
Applying the classifying space functor yields a homotopy pull-back square
BG(ĝ, n; τ̃1, . . . , τ̃n) //
��
BMaps0(Σ(ĝ, n), U2)
Bπ
��
n∏
i=1
BLU τ̃i2
Bι //
n∏
i=1
BL0U2.
(5.2)
Our strategy is to calculate H∗(BG(ĝ, n; τ̃1, . . . , τ̃n)) using the Eilenberg–Moore spectral se-
quence of diagram (5.2). We first need to understand the induced cohomology morphisms shown
below
H∗(BMaps0(Σ(ĝ, n), U2))
n⊗
i=1
H∗(BLU τ̃i2 )
n⊗
i=1
H∗(BL0U2).
Bι∗oo
Bπ∗
OO
(5.3)
Part of this was already calculated in [2].
Lemma 5.2. The map Bπ∗ fits into a commutative diagram
A⊗ Λ(b1,...,bn)
(b1+···+bn) ⊗ S(u,w)
∼= // H∗(BMaps0(Σ(ĝ, n), U2))
n⊗
i=1
Λ(bi)⊗ S(ui, wi)
f
OO
∼= //
n⊗
i=1
H∗(BL0U2),
Bπ∗
OO
where deg(bi) = 3, deg(ui) = deg(u) = 2, deg(wi) = deg(w) = 4, A is an exterior algebra with
Poincaré series Pt(A) = (1 + t)2ĝ(1 + t3)2ĝ, and f(bi) = bi, f(ui) = u, f(wi) = w.
14 T.J. Baird
Proof. This is proven in Lemma 4.4 of [2] for Z2-coefficients, but the proof actually works
over Z and hence holds for any coefficient field. �
The Koszul–Tate complex for the homomorphism Bπ∗ is identified with the bigraded complex
(K∗,∗, δ) where
K∗,∗ := Λ(b1, . . . , bn, x1, . . . , xn−1, y1, . . . , yn−1)⊗ S(u1, w1, . . . , un, wn)⊗ Γ(z)⊗A, (5.4)
where Γ(z) is the divided power algebra generated by z, with bi-degrees and differential δ : Ki,j→
Ki+1,j described in the table below.
generator bi-degree δ-derivative
ui (0, 2) 0
bi (0, 3) 0
wi (0, 4) 0
xi (−1, 2) ui − un
yi (−1, 4) wi − wn
z (−1, 3) b1 + · · ·+ bn
Note in particular that K∗,∗ is a free module over
⊗n
i=1 Λ(bi)⊗S(ui, wi) and the cohomology
of (K∗,∗, δ) is isomorphic to H∗(BMaps0(Σ(ĝ, n), U2)).
5.2 Real loop groups
Let S1 = R/2πZ. Denote by LU2 the group of continuous maps from S1 into the unitary
group U2 and by L0U2 the subgroup of LU2 consisting of maps homotopic to a constant map.
The real loop groups LU τ̃i2 appearing in diagram (5.3) arise as subgroups of L0U2. There are
three kinds.
(a) LU τ̃a2 = LO2 sitting inside LU2 in the standard way.
(b) LU τ̃b2 the gauge group of a rank two Möbius bundle over S1.
(c) LU τ̃c2 = {g : S1 → U2|g(θ) = g(θ + π)} where the bar means entry-wise complex conjuga-
tion.
There is one real loop group of type (a) for each real component of Στ over which E τ̃ is
trivial, one of type (b) for each real component for which E τ̃ is nonorientable, and a positive
number of type (c) if and only if Σ \ Στ is connected.
Lemma 5.3. Over coefficient fields k of characteristic 6= 2 we have an isomorphism
H∗(BLU τ̃2 ; k) =
{
Λ(q)⊗ S(p) if τ̃ = τ̃a or τ̃c,
k if τ̃ = τ̃b,
where deg(q) = 3 and deg(p) = 4. In the prior case, the restriction map
H∗(BL0U2) //
∼=
��
H∗(BLU τ̃2 )
∼=
��
Λ(b)⊗ S(u,w) // Λ(q)⊗ S(p)
sends b to q, w to p, and u to 0.
Cohomology of the Moduli Space of Rank Two 15
Proof. (a) We start with the case LU τ̃a2 = LO2 which is surely well known (the two-fold cover
BLSO2 → BLO2 reduces the problem to the case of LSO2 = LU1, 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 BLO2
∼= Maps0(S1, BO2) with the function space of maps homotopic to
a constant map. Consider the evaluation map
ev : S1 ×Maps0
(
S1, BO2
)
→ BO2.
If p1 ∈ H4(BO2; k) denotes the first Pontryagin class, then we have an isomorphism H∗(BLO2)
∼= Λ(q) ⊗ S(p) where p =
∫
[pt] ev∗(p1) ∈ H4(BLO2; k) and q =
∫
[S1] ev∗(p1) ∈ H3(BLO2; k)
where
∫
denotes slant product with respect to the homology classes [pt], [S1].
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
LσG := {g : I → G | g(0) = σ(g(1))}.
(b) We have LU τ̃b2
∼= LσO2 where σ is an orientation reversing orthogonal change of basis.
Note that σ restricts to an automorphism of SO2. It was proven in [3, Proposition 7.6] that for
coefficient fields of characteristic other than two,
H∗(BLσSO2) ∼= H∗(BLSO1) ∼= H∗(BZ2) ∼= k.
Since LσSO2 ⊂ LσO2 is a subgroup of index two, we deduce that H∗(BLσO2) ∼= H∗(BLσSO2)Z2
is the subring of H∗(BLσSO2) invariant under the residual action by Z2
∼= LσO2/LσSO2. In
particular, H∗(BLσO2) ∼= kZ2 ∼= k.
(c) Consider the map I ↪→ S1 that embeds I as a half circle. Then restriction determines an
isomorphism
LU τ̃c2
∼= LσU2, (5.5)
where σ is entry-wise complex conjugation. By [3, Corollary 7.5], the inclusions LO2 ⊂ LσU2
induces an isomorphism
H∗(LσU2) ∼= H∗(BLO2) ∼= Λ(q)⊗ S(p).
Because (5.5) is induced by including I as a half circle in S1, we have a commutative diagram
H∗(BLO2) H∗(BLO2)
f∗
oo
H∗(BLU τ̃c2 )
∼=
OO
H∗(BLU2)
i∗
oo
OO
where the vertical arrows are induced by inclusion and f∗ is induced by a 2-fold covering map
f : S1 → S1. Functoriality properties of the slant product imply that
f∗(p) =
∫
f∗[pt]
ev∗(p1)) =
∫
[pt]
ev∗(p1)) = p,
f∗(q) =
∫
f∗[S1]
ev∗(p1)) =
∫
2[S1]
ev∗(p1)) = 2q.
Since 2 is invertible, we can simply relabel 2q as q as an element of H∗(BLU τ̃c2 ) completing the
proof. �
16 T.J. Baird
5.3 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.
Proof of Theorem 5.1. DenoteBG τ̃ = BG(ĝ, n; τ̃1, . . . , τ̃n) from the homotopy pull-back diag-
ram (5.3). The associated Eilenberg–Moore spectral sequence EM∗,∗r converges to H∗(BG τ̃ ).
The second page EM∗,∗2 , equals the cohomology of the bi-graded differential graded algebra
(K∗,∗ ⊗R∗ M∗, δ ⊗ 1) where
• (K∗,∗, δ) is the Koszul–Tate complex (5.4),
• M∗ = M0,∗ :=
⊗n
i=1H
∗(BLU τ̃i2 ), and
• R∗ = R0,∗ :=
⊗n
i=1H
∗(BL0U2) =
⊗n
i=1 Λ(bi)⊗ S(ui, wi).
Suppose that there are a ≤ n real circles above which the real bundle is nonorientable and set
b = n− a. Then by Lemma 5.3 we have
n⊗
i=1
H∗(BLU τ̃i2 ) ∼= ⊗bi=1Λ(qi)⊗ S(pi)
and
K∗,∗ ⊗R∗ M∗ ∼= Λ(q1, . . . , qb, x1, . . . , xn−1, y1, . . . , yn−1)⊗ S(p1, . . . , pb)⊗ Γ(z)⊗A,
which comes with differential δ′ = δ ⊗ 1 satisfying δ′(qi) = δ′(pi) = δ′(xi) = δ′(A) = 0, and
δ′(yi) =
pi − pn if b = n,
pi if b < n and i ≤ b,
0 if b < n and i > b,
δ(z) =
{
q1 + · · ·+ qb if b > 0,
0 if b = 0.
The cohomology EM∗,∗2 = H(K∗,∗ ⊗R∗ M∗, δ′) is easily calculated in all cases and can be
expressed in the original variables by abuse of notation.
In case 0 < b < n, we have
EM∗,∗2
∼=
Λ(q1, . . . , qb)
(q1 + · · ·+ qb)
⊗ Λ(x1, . . . , xn−1, yb+1, . . . , yn−1)⊗A.
In case b = n (i.e., the real bundle restricts to a trivial bundle over all real circles), we have
EM∗,∗2
∼=
Λ(q1, . . . , qn)
(q1 + · · ·+ qn)
⊗ Λ(x1, . . . , xn−1)⊗ S(p)⊗A,
where p is represented by cocycle p1. In case b = 0 (i.e., the real circle are separating and the
real bundle restricts to nonorientable bundles for all of them), then δ′ = 0 and we have
EM∗,∗2
∼= Λ(x1, . . . , xn−1, y1, . . . , yn−1)⊗ Γ(z)⊗A.
If b > 0, then the bigraded algebra EM∗,∗2 is generated by homogenous elements lying in
EM0,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 graded-
commutative algebra and an associated graded algebra of H∗(BG τ̃ ), we deduce that H∗(BG τ̃ )
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. �
Cohomology of the Moduli Space of Rank Two 17
6 Proofs of Theorem 1.2 and Corollary 1.3
Proof of Theorem 1.2. For the generic case, simply combine Theorems 1.1 and 5.1, and
Proposition 3.1.
It remains to show that H∗(M(E, τ̃); k) is not an exterior algebra if (Σ, τ) is type I of even
genus greater than two, E has odd degree, and E τ̃ is nonorientable on all components of Στ .
According to Theorem 5.1, the cohomology ring H∗(BG τ̃ ) has Poincaré series
Pt
(
BG τ̃
)
= (1 + t)g
(
1 + t3
)g
/
(
1− t2
)
. (6.1)
According to Proposition 3.1, the normal bundles of all strata are orientable, so the Thom
spaces satisfy the Thom isomorphism. Since the unstable strata all have codimension greater
than (g− 1) (see Section 2.3), then consideration of Thom–Gysin sequences yields equalities for
Betti numbers:
dim(H i(M(E, τ̃))) = dim(H i(BG τ̃ )), for i ≤ g − 2.
In particular, if g ≥ 4 then the first and second Betti numbers of M(E, τ̃) and BG τ̃ must agree.
From (6.1), the first Betti number of BG τ̃ is g and the second is
(
g
2
)
+ 1 which implies that
H∗(M(E, τ̃); k) cannot possibly be an exterior algebra. �
Lastly, we explain the relationship between the cohomology of M(r, d, τ̃) and that of the
fixed determinant moduli space MΛ(r, d, τ̃). This is analogous to the complex version described
in Atiyah–Bott [1, p. 578].
Consider the trivial C∞-line bundle Σ× C, with trivial real structure
c̄(x, z) = (τ(x), z̄).
The moduli space M(1, 0, c̄) is a group isomorphic to (S1)g. The subgroup Tr ≤ M(1, 0, c̄) of
r-th roots of unity acts by tensor product on MΛ(r, d, τ̃). The tensor product map MΛ(r, d, τ̃)×
M(1, 0, c̄) → M(r, d, τ̃) is a covering space with transitive deck transformation group Tr, so it
descends to a diffeomorphism
M(r, d, τ̃) ∼= MΛ(r, d, τ̃)×Tr M(1, 0, c̄),
where we have taken the mixed quotient with respect to the tensor product action of Tr on
MΛ(r, d, τ̃) and M(1, 0, c̄). Since Tr acts by translations on M(1, 0, c̄), it acts trivially on coho-
mology. It follows that for characteristic relatively prime to r, we have a ring isomorphism
H∗(M(r, d, τ̃); k) ∼= H∗(MΛ(r, d, τ̃); k)Tr ⊗H∗(M(1, 0, c̄); k).
Finally, M(1, 0, c̄) is diffeomorphic to (S1)g, so H∗(M(1, 0, c̄); k) is an exterior algebra with g
generators in degree one. Corollary 1.3 follows.
Acknowledgements
A special thanks to Jacques Hurtubise and Ben Smith who began this project as collaborators
and contributed to some of the exposition. Jacques in particular helped motivate this project
by establishing criteria for the normal bundles of the real Harder–Narasimhan stratification to
be non-orientable (a proof later superseded by the work of Okonek–Teleman). Thanks also to
Andrei Teleman and other the participants at the Real vector bundles conference in Brest for
helpful discussions, and to the referees for helpful comments. This research was supported by
an NSERC Discovery Grant.
18 T.J. Baird
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http://arxiv.org/abs/1003.5285
1 Introduction
2 Preliminaries
2.1 Topological classifications
2.2 The complex Harder–Narasimhan stratification
2.3 The real Harder–Narasimhan stratification
3 Orientability of normal bundles
4 Proof of Theorem 1.1
5 Real gauge groups
5.1 Constructing the classifying space
5.2 Real loop groups
5.3 The spectral sequence
6 Proofs of Theorem 1.2 and Corollary 1.3
References
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