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Advances in Mathematics 185 (2004) 370–399

http://www.elsevier.com/locate/aim

The mod 2 cohomology of ﬁxed point sets ofanti-symplectic involutions
Daniel Biss,a Victor W. Guillemin,a and Tara S. Holmb,
aDepartment of Mathematics, Massachusetts Institute of Technology 2-488, Cambridge, MA 02139, USAb
Department of Mathematics, No. 3840, University of California, Berkeley 94720-3840, USA
Received 27 February 2002; accepted 3 July 2003
Communicated by D.S. Freed

Abstract
Let M be a compact, connected symplectic manifold with a Hamiltonian action of acompact n-dimensional torus G ¼ T n : Suppose that s is an anti-symplectic involution
compatible with the G-action. The real locus of M is X ; the ﬁxed point set of s: Duistermaat
uses Morse theory to give a description of the ordinary cohomology of X in terms of thecohomology of M: There is a residual GR ¼ ðZ=2ZÞn action on X ; and we can use
Duistermaat’s result, as well as some general facts about equivariant cohomology, to prove an
equivariant analogue to Duistermaat’s theorem. In some cases, we can also extend theorems ofGoresky–Kottwitz–MacPherson and Goldin–Holm to the real locus.
r 2003 Elsevier Inc. All rights reserved.
Keywords: Symplectic manifolds; Equivariant cohomology; Real locus

1. Introduction
Atiyah observed in [A] that if M is a compact symplectic manifold and t a
Hamiltonian action of an n-dimensional torus G on M; then the cohomology groupsof M can be computed from the cohomology groups of the ﬁxed point set M G of t:
Explicitly, if F : M-T is the moment map for t; then a generic component Fx of FCorresponding author. Fax: +1-617-253-4358.
E-mail addresses: daniel@math.mit.edu (D. Biss), vwg@math.mit.edu (V.W. Guillemin), tsh@math.
berkeley.edu (T.S. Holm).
0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.aim.2003.07.007

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is a perfect Bott–Morse function. Using Fx ; we may computeH  ðM; RÞ ¼

NX

H di ðFi ; RÞ;

ð1:1Þ

i¼1

where the Fi are the connected components of M G and di is the Bott–Morse index of
Fi : This result is also true in equivariant cohomology [AB,BV]:
HG ðM; RÞ ¼

N
X

HGdi ðFi ; RÞ ¼

i¼1

N
X

H di ðFi  BG; RÞ:

ð1:2Þ

i¼1

This is a consequence of Atiyah’s result and Kirwan’s equivariant formality theorem
for Hamiltonian G-manifolds, as shown in [Ki].
In [Du], Duistermaat proved a ‘‘real form’’ version of (1.1). Let s : M-M be ananti-symplectic involution with the property thats3tg ¼ tg1 3s

ð1:3Þ

and let X ¼ M s be the ﬁxed point set of s: We call X the real locus of M: The
motivating example of this setup is a complex manifold M with a complex
conjugation s: Duistermaat proved thatH  ðX ; Z=2ZÞ ¼

Xdi

H  2 ðFis ; Z=2ZÞ;

ð1:4Þ

iAI

where IDf1; y; Ng is the set for which s preserves Fi : The Z=2Z coefﬁcients are
essential here; the theorem does not hold with real coefﬁcients. (See the comments at
the end of this section.)
The ﬁrst of the four main theorems of this paper is an equivariant analogue of(1.4) similar to the equivariant analogue (1.2) of Atiyah’s result (1.1). By (1.3), thegroup
GR ¼ fgAG j g2 ¼ idgDðZ=2ZÞn

ð1:5Þ

acts on X and we will prove the following theorem in Section 2.
Theorem A. Suppose M is a symplectic manifold with a Hamiltonian action t of atorus T n ¼ G and an anti-symplectic involution s: Let X ¼ M s denote the real locus of
M: Then the group GR acts on X ; and the GR -equivariant cohomology of X with Z=2Z
coefficients is
HG R ðX ; Z=2ZÞ ¼

X

d
 i

HGR 2 ðFis ; Z=2ZÞ:iAIAs above, the subset IDf1; y; Ng is the set for which s preserves Fi :

ð1:6Þ

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The idea of the proof will be to derive (1.6) from (1.4) by a simple trick.
The second of the main theorems concerns the structure of
HG R ¼ ðX ; Z=2ZÞ
as a module over the ring
HG R ¼ HG R ð ptÞ ¼ Z=2Z½x1 ; y; xn :

ð1:7ÞLet BGR and EGR denote the classifying space and classifying bundle of GR : Then by
the Borel deﬁnition of equivariant cohomology
HG R ðX ; Z=2ZÞ ¼ H  ðX GR EGR ; Z=2ZÞ:The cohomology on the right-hand side can be computed by the spectral sequence
associated with the ﬁbration
X GR EGR -BGR ;and we will deduce from this computation the following theorem.
Theorem B. The equivariant cohomology HG R ðX ; Z=2ZÞ is a free module over HG R
generated in dimension zero. Moreover, as an HG R module, HG R ðX ; Z=2ZÞ isisomorphic to
HG R #Z=2Z H  ðX ; Z=2ZÞ:ð1:8Þ

The idea of the proof is to show that this spectral sequence collapses at its E2 term,i.e., X is equivariantly formal. In particular, this implies that HG R ðX Þ is a freemodule over Z=2Z½x1 ; y; xn :Isomorphisms (1.1), (1.2), (1.4), and (1.6) are all isomorphisms in additive
cohomology. The next two sections of this paper will be concerned with the ring
structure of HG R ðX ; Z=2ZÞ: Henceforth, we will assume that the ﬁxed point set M G is
ﬁnite and that the one-skeleton of t;
M ð1Þ ¼ fpAM j dimðG pÞp1g;
is of dimension 2: It is not hard to see that these two assumptions imply thatM ð1Þ ¼

N[Ei ;Ei DCP1 :i¼1

Moreover, Ei is point-wise ﬁxed by an ðn  1Þ-dimensional torus Hi ; and thediffeomorphism
Ei -CP1

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intertwines the action of G=Hi on Ei with the standard S1 action on CP1 : Inparticular, Ei contains exactly two G-ﬁxed points. Thus, one can describe theintersection properties of the Ei by a graph G with edges ei corresponding to thespheres Ei and vertices VG ¼ M G : Two vertices p and q are joined by the edge ei if
EiG ¼ fp; qg: Furthermore, each edge, ei is labeled by a weight a* ei of G; the weightassociated with the intertwining homomorphism
G=Hi -S1 :Let i : M G -M be the inclusion of the ﬁxed points into the manifold, and consider
the induced map in equivariant cohomology,i : HG ðM; CÞ-HG ðM G ; CÞ:ð1:9Þ

By a theorem of Kirwan [Ki], this map is injective. Moreover,
HG ðM G ; CÞ ¼

M

HG ðfpg; CÞ;ð1:10Þ

pAM G

and since
HG ðf pg; CÞDSðg Þ;
one can regard an element of HG ðM G ; CÞ as a map
f : M G -Sðg Þ:

ð1:11Þ

Goresky et al. [GKM] computed the image of i ; thus determining not just theadditive equivariant cohomology of M; but in fact the ring structure of thiscohomology.
Theorem 1.1 (Goresky et al. [GKM]). A map f : M G -Sðg Þ is in the image of i if
and only if for each edge ei ¼ fp; qg of Gf ð pÞ  f ðqÞA*ae1 Sðg Þ:

ð1:12Þ

The third of the four main theorems of this paper will be a Z=2Z version of thisresult for the manifold X : We deﬁne the one-skeleton of the real locus to be the set
X ð1Þ ¼ fxAX j #ðGR xÞp2g:

ð1:13Þ

Assume in addition to the above that M G ¼ X GR and the real locus of the oneskeleton is the same as the one-skeleton of the real locus. We will call a manifold withthese properties a mod 2 GKM manifold. The mapi : HG R ðX ; Z=2ZÞ-HG R ðX GR ; Z=2ZÞ

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D. Biss et al. / Advances in Mathematics 185 (2004) 370–399

is injective, and by factoring (1.14) through the map
HG R ðX ; Z=2ZÞ-HG R ðEis ; Z=2ZÞ;ð1:15Þ

where Eis ¼ Ei -X DRP1 ; we obtain the following theorem.
Theorem C. Suppose M is a mod 2 GKM manifold. An elementf AHG R ðX GR ; Z=2ZÞ
can be thought of as a map f : VG -Z=2Z½x1 ; y; xn ; and such a map f is in the imageof i if and only if, for each edge e ¼ fp; qg of G
fp  fq Aae Z=2Z½x1 ; y; xn ;where ae AZ=2Z½x1 ; y; xn is the image of the weight a* e :This completely determines the ring structure of HG R ðX ; Z=2ZÞ: This theorem isproved independently by Schmid [S] using different techniques. The main examples
of mod 2 GKM manifolds include the real loci of non-singular projective toricvarieties and real loci of coadjoint orbits, including Grassmannian and ﬂag varieties.
In the case of toric varieties, this equivariant cohomology ring was computed alreadyby Davis and Januszkiewicz [DJ], but our description is quite different from theirs.
In [GH], Goldin and Holm generalize the GKM result to the case where the oneskeleton has dimension at most 4: Assume in addition to the dimension hypothesisthat M G ¼ X GR and the real locus of the one-skeleton is the same as the one-skeletonof the real locus. We will call a manifold with these properties a mod 2 GH manifold.
The last of the main theorems is a Z=2Z version of the result of Goldin and Holm forthe real locus X : For a subgroup HR of GR ; we will let pHR will denote the change of
coefﬁcient map
pHR : HG R -HH
R

associated with the inclusion HR +GR : In Section 6, we will prove the following
theorem.
Theorem D. Suppose that M is a mod 2 GH manifold with G fixed points fixed points
M G ¼ f p1 ; y; pd g: Let fi AHG R denote the restriction of f AHG R ðX Þ to the fixed pointpi : The image of the injection i : HG R ðX Þ-HG R ðX GR Þ is the subalgebra of functions
ð f1 ; y; fd ÞA"di¼1 HG R which satisfy
8 
GR

>
< pHR ðfij Þ ¼ pHR ð fik Þ if f pi1 ; y; pil g ¼ ZHR ;
l
fij
P
GR

>
if f pi1 ; y; pil g ¼ ZH
and dim ZHR ¼ 2:
ij ij AHGR
R
j¼1 a1 a2

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for all subgroups HR of GR of order jHR j ¼ 2n1 and all connected components ZHR ofii
X HR ; where a1j and a2j are the (linearly dependent) weights of the GR action on Tpij ZHR :We re-emphasize that Duistermaat’s techniques only apply to additive cohomology. Since we are able to obtain results concerning the ring structure of theequivariant cohomology and its relationship to ordinary cohomology, we also obtainstatements about the ring structure of the ordinary cohomology as well. Indeed, inmany cases, Duistermaat’s isomorphism (1.4) turns out to give a ring isomorphism.
(See Corollaries 5.7 and 5.8 to Theorem C and Corollaries 6.7 and 6.8 to Theorem
D.) When describing these ring isomorphisms, we will make use of the following
notation. The symbol
H 2 ðM; Z=2ZÞ
will denote the subringM

H 2i ðM; Z=2ZÞDH  ðM; Z=2ZÞ;iendowed with a new grading wherein a class in H 2i ðM; Z=2ZÞ is given degree i (andsimilarly for equivariant cohomology). Then under suitable hypotheses, the additive
isomorphism of Duistermaat becomes an isomorphism of graded rings.
In Section 7, we discuss an application of our main theorems to string theory. The
Z=2Z-equivariant cohomology ring of T n with Z=2Z coefﬁcients classiﬁes all possible
orientifold conﬁgurations of Type II string theories, compactiﬁed on T n : We explain
how to compute this cohomolgy ring.
The last section of the paper contains some applications of these results toelementary problems in combinatorics. A typical such application is the following.
Let G be the permutahedron, the Cayley graph of the symmetric group Sn with edges
generated by transpositions. By deﬁnition, the vertices of G are elements of Sn andtwo vertices s and t are joined by an edge if ts1 is a transposition. Our goal is toattach to each vertex s a subset Ss of f1; y; ng such that, for all pairs s and t ofadjacent vertices, either Ss ¼ St or the symmetric differenceðSs  St Þ,ðSt  Ss Þ
is fi; jg; where ts1 ¼ ðijÞ:Let Fnþ1 be the real ﬂag variety in n þ 1 dimensions. We will prove that the set ofsolutions to this problem can be identiﬁed with the set
HG R ðFnþ1 ; Z=2ZÞ;where GR ¼ Z=2Z  ?  Z=2Z is the n-fold product of Z=2Z: The results in thissection were inspired by a remark of Ethan Bolker, who pointed out to us that Z=2Z
representation theory is simply Boolean algebra.

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We conclude these prefatory remarks with a few comments about the Z=2Z
coefﬁcients. We recall Witten’s recipe for computing the homology of compact
manifolds by Morse theory. Let X be a compact manifold, f : X -R a Morsefunction, and Cfi the index i critical set of f : Let Ci be the vector space
M

Ci ¼

pi R

pi ACfiwith basis Cfi :Equip X with a Riemannian metric and let v be the gradient vector ﬁeld of f : For
generic metrics, the stable and unstable manifolds of v intersect transversally. In
particular, for every critical point pACfi ; there are a ﬁnite number of gradient curves
joining p to critical points
q1 ; y; qm ACfi1 :ð1:16Þ

Moreover, each of these points can be assigned an intrinsic orientation
eð p; qj ÞAf71g: Now deﬁne a boundary operator@ : Ci -Ci1
by setting@p ¼

Xeð p; qj Þ qj :Witten [W] was the ﬁrst to explicitly formulate Morse theory in this way; he showedthat @ is a boundary operator, namely @ 2 ¼ 0; and that H ðX ; RÞ is the homology of
the complex ðC; @Þ:In particular, when all the critical points are of even index, @ is automatically zero.
Thus, one gets(dimðHi ðX ; RÞÞ ¼

0;

i odd;#f pACfi g;i even:ð1:17Þ

This fact is key to Atiyah’s result (1.1). He observes that if #M G oN and if f is a
generic component of the moment map, then f is a Morse function with criticalpoints all of even index, so (1.1) is a special case of (1.17).
This recipe for computing homology also works in characteristic two; however,when Z=2Z symmetries are present, the gradient curves joining the points pACfi tothe points on list (1.16) often occur in pairs. For instance, for manifolds which satisfy
the GKM hypotheses, these pairs of curves correspond to the edges ei of the graph G:
That is, each Eis DRP1 in (1.15) contains of a pair of gradient curves joining the twovertices of ei : Hence, the mod 2 version of @ is identically zero.

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One can obtain Duistermaat’s result by exploiting this phenomenon. The goal ofthis article is to push these ideas forward by systematically applying these techniques
to the equivariant setting.

2. The equivariant cohomology of the real locus
Recall M 2d is a symplectic manifold with a Hamiltonian action t of a torus
G ¼ T n : Suppose further that there is an anti-symplectic involution s : M-M withthe property thats3tg ¼ tg1 3s:
Let X ¼ M s be the ﬁxed point set of s: We call X the real locus of M: Recall that
Duistermaat proved the following equality, computing the ordinary cohomology of
the real locus.
H  ðX ; Z=2ZÞ ¼

Xdi

H  2 ðFis ; Z=2ZÞ;

ð2:1Þ

iAI

where IDf1; y; Ng is the set for which s preserves Fi ; and the di are the indices of
the ﬁxed point sets Fi :We will prove the equivariant analogue (1.6) to this equality, computing theadditive structure of the equivariant cohomology:
HG R ðX ; Z=2ZÞ ¼

X

d
 i

HGR 2 ðFis ; Z=2ZÞ:iAI

Proof of Theorem A. Consider the product action of T n on
M  ðCd  ?  Cd Þ
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
nin which each S1 factor acts by multiplication on the corresponding factor of Cd :This is a Hamiltonian action. If ðf1 ; y; fn Þ ¼ F : M-Rn is the moment map
associated with t; then the moment map of this product action is C ¼ ðc1 ; y; cn Þ;withci ðm; z1;1 ; y; z1;d ; y; zd;d Þ ¼ fi ðmÞ þdX

jzi;j j2 :j¼1

Let a ¼ ða1 ; y; an ÞARn : If ai 4 supðfi Þ for every i; then C1 ðaÞ and M  S 2d1 ?  S 2d1 are equivariantly diffeomorphic, so the reduced space
Mred ¼ M==a T n ¼ c1 ðaÞ=T n

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is diffeomorphic to M T n ðS2d1  ?  S 2d1 Þ: Moreover, there is another actionof T n on M  Cd  ?  Cd ; namely t coupled with the trivial action on ðCd Þn : Since
this commutes with the product action, it induces a Hamiltonian action of T n on
Mred : In addition, one gets from s an involution
ðm; z1 ; y; zd Þ/ðsðmÞ; z1 ; y; zd Þof M  Cd  ?  Cd : This induces an anti-symplectic involution s* on Mred : Thus,
one can apply Duistermaat’s theorem to Mred to get a formula for the cohomology ofthe spaces*
Mred¼ X GR ðSd1  ?  S d1 Þin terms of the cohomology of the spaces
Zid :¼ Fis GR ðS d1  ?  Sd1 Þ ¼ Fis  ðRP d1  ?  RP d1 Þ:
Now Fis  BGR is obtained from Zid by attaching cells of dimension d and higher.
So, for ﬁxed k; the sequence H k ðZid ; Z=2ZÞ stabilizes as d grows large, and moreoveris equal to the equivariant cohomology of X : Thus one obtains from (2.1) the
following real analogue:
X di
HG R ðX ; Z=2ZÞ ¼
HGR 2 ðFis ; Z=2ZÞ;where GR ¼ Z=2Z  ?  Z=2Z:&

3. A spectral sequence
The goal of this section is to determine the structure of HG R ðX ; Z=2ZÞ as a module
over the ring HG R ¼ Z=2Z½x1 ; y; xn : We do this by calculating the E2 -term of theLeray–Serre spectral sequence converging to HG R ðX ; Z=2ZÞ and deducing, bydimensional considerations obtained in the previous section, that the spectral
sequence must collapse. This then gives us the desired statement about the EN -termand hence HG R ðX ; Z=2ZÞ:
Recall that by deﬁnition, we have
HG R ðX ; Z=2ZÞ ¼ H  ðX GR EGR ; Z=2ZÞ;where EGR is the total space of the universal GR -bundle. Denote this ﬁber product by
E: We then have a ﬁbration p : E-BGR with ﬁber X : Let H X denote the local
coefﬁcient system on BGR associated to this ﬁbration. The E2 -term we would like tocompute is thenE2 ¼ H  ðBGR ; H X Þ:

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The computation takes place in two steps. The ﬁrst step, which is the technical heartof the argument, consists of carrying out the computation in the one-dimensionalcase. The remainder of the proof consists of a relatively straightforward exercise inbookkeeping.
Lemma 3.1. Let GR ¼ Z=2Z; so that BGR ¼ KðZ=2Z; 1Þ ¼ RPN : Then
H  ðBGR ; H X Þ is generated over HG R ¼ ðZ=2ZÞ½x in degree zero by H  ðX ÞGR :Moreover, the only relation is given by x ða þ nðaÞÞ ¼ 0 for aAH  ðX Þ and nAGR :Proof. By deﬁnition, the cohomology H  ðBGR ; H X Þ that we would like tocompute is the group cohomology H  ðGR ; H  ðX ÞÞ with respect to the naturalaction of GR on H  ðX Þ; we will henceforth pass back and forth between these
two notations without comment. Our goal is to understand the cohomology
as a module over the cohomology ring H  ðGR ; Z=2ZÞ corresponding to the trivial
action of GR on Z=2Z:
Denote the non-trivial element of Z=2Z by n: Consider the GR -moduleH  ðX Þ"H  ðX Þ with GR -action deﬁned by the equation nða; bÞ ¼ ðnðbÞ; nðaÞÞ forall a; bAH  ðX Þ: We then get a short exact sequence of GR -modulesfg

0-H  ðX Þ ! H  ðX Þ"H  ðX Þ ! H  ðX Þ-0;

ð3:1Þ

where f ðaÞ ¼ ða; aÞ and gða; bÞ ¼ a þ b: Of course, it is completely essential to thisidentiﬁcation of the cokernel of f that we work over Z=2Z; in order for this sequence
to be exact, we must have gða; bÞ ¼ a  b; but this would not be a map of Z=2Zmodules if we did not also have gða; bÞ ¼ a þ b:
We would like to consider the long exact cohomology sequence associated with
(3.1). Observe that we have isomorphisms
H  ðX Þ"H  ðX ÞD Z=2Z½GR #Z=2Z H  ðX Þ
D Z½GR #Z H  ðX Þ
D HomZ ðZ½GR ; H  ðX ÞÞof GR -modules. The ﬁrst isomorphism is more or less the deﬁnition of the left-handside, the second follows from the fact that H  ðX Þ is 2-torsion, and the third follows
from the ﬁniteness of GR : Thus, the module H  ðX Þ"H  ðX Þ is co-induced and its
higher cohomology vanishes. Moreover, the map a/ða; nðaÞÞ provides anisomorphism of H  ðX Þ with ðH  ðX Þ"H  ðX ÞÞGR ; so we have
H 0 ðGR ; H  ðX Þ"H  ðX ÞÞDH  ðX Þ

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and, of course, H 0 ðGR ; H  ðX ÞÞ ¼ H  ðX ÞGR : The long exact sequence in question,
therefore, takes the following form:

The ﬁrst map in this sequence is the natural inclusion, and the second sends a toa þ nðaÞ: Therefore, H 1 ðBGR ; H X Þ is the sought-after quotient of H  ðX ÞGR by the
subgroup of all elements of the form a þ nðaÞ; as a result, so is each H n ðBGR ; H X Þ
for nX1: Recall that H  ðBGR Þ ¼ ðZ=2ZÞ½x where x is a class of degree 1. The
desired result then follows from the fact that the connecting homomorphisms are
multiplication by x: &The remainder of the game consists in playing the results of Lemma 3.1 andTheorem A off of one another.
Corollary 3.2. The action of GR on H  ðX Þ is trivial. Thus, in the case that GR ¼ Z=2Z;
Theorem B holds.
Proof. Lemma 3.1 computes the E2 -term of the Leray–Serre spectral sequence
converging to HG R ðX ; Z=2ZÞ in the case GR ¼ Z=2Z: Now, the dimensions of thegraded pieces of this E2 -term are maximized precisely when GR acts trivially on
H  ðX Þ: Moreover, the results of the previous section tell us that the graded pieces of
HG R ðX ; Z=2ZÞ have exactly these maximal dimensions. Since the EN -term of aspectral sequence can only be as large as its E2 -term, this tells us that the action mustbe trivial and further that
HG R ðX ; Z=2ZÞ ¼ EN ¼ E2 :In this case, H  ðX ÞGR ¼ H  ðX Þ; and the relation x ða þ nðaÞÞ ¼ 0 is automaticallysatisﬁed, so Lemma 3.1 tells us that H  ðBGR ; H X Þ is a free module over Z=2Z½xgenerated in degree zero by H  ðX Þ: This completes the proof of Theorem B in thecase GR ¼ Z=2Z:
The triviality of the action of GR on H  ðX Þ for higher-dimensional GR follows by
restricting to arbitrary one-dimensional subtori. &The fact that GR acts trivially on H  ðX Þ is not new; it can also be derived from
Duistermaat’s original argument. Indeed, Duistermaat’s isomorphism can be seen to
be GR -equivariant, and the result then follows from the connectedness of the torus.
Using this fact would have somewhat simpliﬁed our argument, but we chose to give

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the above proof so as to avoid appealing to unpublished modiﬁcations of theliterature.
We now have the technical input to handle the general case.
Lemma 3.3. Let GR DðZ=2ZÞn ; for any positive integer n: Then H  ðBGR ; H X Þ is afree module over HG R ¼ ðZ=2ZÞ½x1 ; y; xn generated by H 0 ðBGR ; HG R ÞDH  ðX Þ:
Proof. This is now completely classical. By Corollary 3.2, the action of GR on H  ðX Þis trivial, so that H X is actually the constant sheaf H  ðX Þ: Therefore, we haveH  ðBGR ; H X Þ ¼ H  ðBGR ; Z=2ZÞ#Z=2Z H  ðX Þ ¼ Z=2Z½x1 ; y; xn #Z=2Z H  ðX Þ:This completes the proof.

&

Our goal is now entirely within reach; we need only combine the results we have
proven so far as in the proof of Corollary 3.2 to establish the HG R -module
isomorphism
HG R ðX ; Z=2ZÞDHG R #Z=2Z H  ðX ; Z=2ZÞ:ð3:2Þ

Proof of Theorem B. Lemma 3.3 tells us that the E2 -term of the Leray–Serre spectral
sequence converging to HG R ðX ; Z=2ZÞ takes precisely the form that HG R ðX ; Z=2ZÞ
itself is asserted to have. However, the results of the previous section tell us that thegraded pieces of HG R ðX ; Z=2ZÞ have the same dimension as those of this E2 -term,and hence that the spectral sequence collapses. Therefore,
HG R ðX ; Z=2ZÞ ¼ EN ¼ E2 ;so HG R ðX ; Z=2ZÞ is a free module over HG R generated in dimension zero. Thus, its
additive structure is as given by (3.2). &4. Chang–Skjelbred in Z=2Z
As a result of the collapse of the spectral sequence proved in the previous section,
the mapi : HG R ðX ; Z=2ZÞ-HG R ðX GR ; Z=2ZÞ
is an injection. In the case of the original manifold M; the Chang–Skjelbred
theorem [CS] identiﬁes the image of this map. We prove a Z=2Z version of thattheorem here.
As usual, let GR ¼ ðZ=2ZÞn be the n-dimensional ‘‘real torus,’’ M a GR manifold, and M GR the ﬁxed point set of the action; we let i : M GR +M denotethe inclusion.

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Theorem 4.1. Suppose that HG R ðM; Z=2ZÞ is a free HG R -module. For a subgroup
HR oGR ; let iHR : M GR +M HR denote the inclusion. Then we have\iH
HG R ðM HR ; Z=2ZÞ:i HG R ðM; Z=2ZÞ ¼
R
HR oGR
jHR j¼2n1

Our proof closely models the argument given in [GS], with appropriate
modiﬁcations.
First of all, recall that
HG R ¼ H  ððRPN Þn ; Z=2ZÞ ¼ Z=2Z½x1 ; y; xn ;with degðxi Þ ¼ 1: Moreover, we may view each xi as a linear functional
xi : GR -Z=2Z; that is, an element of GR : This allows us to identify HG R with thesymmetric algebra SðGR Þ; a fact analogous to the ordinary identiﬁcation HG DSðg Þ
when G is a torus with Lie algebra g: This allows us to view elements of HG R as
polynomial functions on GR :Lemma 4.2. Let KR oGR be a subgroup and f : M-GR =KR a GR -equivariant map. IfrASðGR Þ annihilates KR ; then it must also kill HG R ðM; Z=2ZÞ:Proof. We have a sequence of GR -equivariant maps
M-GR =KR -pt:

ð4:1Þ

Note that
HG R ðGR =KR Þ ¼ H  ðGR =KR GR EGR ; Z=2ZÞ
¼ H  ðBKR ; Z=2ZÞ
¼ SðKR Þ:
Sequence (4.1) of GR -spaces therefore gives rise to the following diagram of algebras:

Therefore, the map SðGR Þ-HG R ðM; Z=2ZÞ deﬁning the module structure factors asSðGR Þ-SðKR Þ-HG R ðM; Z=2ZÞ;and the proof is complete.

&

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383This observation furnishes us with the fundamental tool in proving localization
theorems for equivariant cohomology.
Proposition 4.3. Let X be a closed GR -invariant submanifold of M: For some positive
integer L; there exist subgroups ðKR Þ1 ; y; ðKR ÞL of G; each of which is an isotropy
subgroup of some point pAM \X ; such that for any a1 ; y; aL AGR with ai jðKR Þi ¼ 0; theproduct a1 a22 ?a2L AHG R kills HG R ðM \X Þ:
Proof. Let U be a GR -invariant tubular neighborhood of X ; it sufﬁces to prove thedesired result for the module HG R ðM \U; Z=2ZÞ: Now, given any orbit Xi in M \U
with isotropy subgroup ðKR Þi ; we may ﬁnd a GR -invariant open neighborhood Ui of
Xi admitting a GR -equivariant map Ui -GR =ðKR Þi : By compactness, we may cover
M \U by ﬁnitely many such sets U1 ; y; UL : We now show by induction that for all
rpL; if a1 ; y; ar are elements of GR with ai jðKR Þi ¼ 0; then a1 a22 ?a2r annihilates
HG R ðU1 ,?,Ur ; Z=2ZÞ:The case r ¼ 1 is simply a restatement of Lemma 4.2. For the inductive step,
consider the Meyer–Vietoris sequence associated to the cover U1 ,?,Ur ¼
ðU1 ,?,Ur1 Þ,Ur : Denoting U1 ,?,Ur1 by V ; we ﬁnd an exact sequenceðV Þ"HGkþ1ðUr Þ:
HGk R ðV -Ur Þ-H kþ1 ðV ,Ur Þ-HGkþ1
R
R
Now, since V -Ur CUr ; we have a GR -equivariant map V -Ur -Xr ; and so the lefthand term of the sequence is killed by ar : Meanwhile, by induction, the right-handterm is annihilated by a1 a22 ?a2r1 ar ; and so the product a1 a22 ?a2r kills the middle
term. &
We will also need a relative version of the same result.
Proposition 4.4. Under the hypotheses of Proposition 4.3, the module HG R ðM; X Þ isannihilated by the element a21 a42 ?a4L AHG R :Proof. Of course, the map of pairs ðM; X Þ-ðM=X ; X Þ is an equivalence, so it
sufﬁces to compute HG R ðM=X ; X Þ: Once again, let U be a GR -equivariant tubularneighborhood of X : We cover M=X by two open sets U=X and M \X ; since, theprojection map U=X -X =X is an equivalence, so we may identify HG R ðM; X Þ withthe kernel of the map HG R ðMÞ-HG R ðUÞ: Now, let us write the Meyer–Vietoris
sequence for this cover:ðU\X Þ-HGk R ðM=X Þ-HGk R ðU=X Þ"HGk R ðM \X Þ:
HGk1
R
But by the above discussion, this gives rise to the following exact sequence:
HGk1
ðU\X Þ-HGk R ðM; X Þ-HGk R ðM \X Þ:
R

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Now, Proposition 4.3 applies to both ends of this sequence, so the middle term iskilled by ða1 a22 ?a2L Þ2 ¼ a21 a42 ?a4L : &Proposition 4.4 gives us the basic localization results we will need.
Corollary 4.5. In the setting of Proposition 4.3, the kernel and cokernel of the mapi : HG R ðM; Z=2ZÞ-HG R ðX ; Z=2ZÞ
are annihilated by the same element a21 a42 ya4L AHG R :Proof. Simply apply Proposition 4.4 to the exact sequence
HGk R ðM; X Þ-HGk R ðMÞ-HGk R ðX Þ-HGkþ1
ðM; X Þ:R&

Corollary 4.6. The kernel of the map i : HG R ðMÞ-HG R ðM GR Þ is torsion, and hence
trivial when HG R ðM; Z=2ZÞ is a free module.
Corollary 4.7. For every subgroup KR oGR ; there exists a monomial p ¼ a1 yaNannihilating the cokernel of the map i : HG R ðMÞ-HG R ðM KR Þ such that no ai vanishes
on KR :
Proof. Let qAM \M KR ; and let KR0 be its isotropy subgroup. Since obviously
KR0 NKR ; there is an a0 : GR -Z=2Z with a0 jK 0 ¼ 0 and a0 jKR a0: But by Corollary 4.5,R
we can ﬁnd a monomial which is a product of elements of the form a0 and annihilates
coker i : &Finally, we are ready to prove our main result.
Proof of Theorem 4.1. First of all, since the map i factors as
iHR

M GR + M HR +M;
we know that for all HR ; the inclusion imði ÞCimðiH
Þ holds.
RFor the other direction, recall ﬁrst that HGR ðM; Z=2ZÞ is free; therefore, by
Corollary 4.6 the map i is injective and we may consequently view HG R ðM; Z=2ZÞ asa submodule of HG R ðM GR ; Z=2ZÞ: Suppose fe1 ; y; ek g is an SðGR Þ-basis for
HG R ðM; Z=2ZÞ: By Corollary 4.7, there is a monomial p ¼ a1 ?aN with
peAHG R ðM; Z=2ZÞ for every eAHG R ðM GR ; Z=2ZÞ: Thus, we may write

pe ¼ f1 e1 þ ? þ fk ek

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for unique fi ASðGR Þ: Now, since SðGR Þ is a unique factorization domain, we may
divide both sides of this identity by p and cancel common factors to obtain theformulae¼

g1
gk
e1 þ ? þ ek ;
p1
pk

ð4:2Þ

where the gi are uniquely determined elements of SðGR Þ and the pi are uniquelydetermined divisors of p such that gi and pi are relatively prime.
Suppose now that e were actually in imðiH
Þ: We may ﬁnd a subset
R
f j1 ; y; jR gCf1; y; Ng such that no aji kills HR ; and q ¼ aj1 ?ajR annihilates thecokernel of the map
HG R ðM; Z=2ZÞ-HG R ðM HR ; Z=2ZÞ:Therefore, multiplying both sides of (4.2) by q; we ﬁnd thataj1 ?ajR e ¼ h1 e1 þ ? þ hk ek
with hi ASðGR Þ: Thus, in (4.2), none of the weights a : G-Z=2Z that divide theÞ for all HR ; then pi ¼ 1 for all i;denominators pi vanish on HR : Hence, if eAimðiH
R
and so (4.2) tells us thate ¼ g1 e1 þ ? þ gk ek AHG R ðM; Z=2ZÞ
and the proof is complete.

&

Now suppose that ZHR is a connected component of M HR for some subgroup HR
of GR of order jHR j ¼ 2n1 : Let iZHR be the inclusion
GR
iZHR : ZH-ZHR
Rof the ﬁxed points of ZHR into ZHR : Let rZHR be the inclusionGR-M GR
rZHR : ZH
Rof the ﬁxed points of ZHR into all of the ﬁxed points. Then, we have the following
corollary of Theorem 4.1.
Corollary 4.8. Suppose that HG R ðM; Z=2ZÞ is a free HG R -module. A classf AHG R ðM GR ; Z=2ZÞ
is in the image of i if and only ifrZH ð f ÞAiZ H ðHG R ðZHR ; Z=2ZÞÞ
R

R

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for every subgroup HR of GR of order jHR j ¼ 2n1 and every connected component ZHRof M HR :Proof. The proof is analogous to the proof of Theorem 1 in [GH]. It follows directlyfrom Theorem 4.1. &5. Real GKM
The goal of this section is to prove an analogue of Theorem 1.1 for the real locus
X of M: The proof will require two hypotheses on X ; namely
X GR ¼ M G

ð5:1Þ

X ð1Þ ¼ X -M ð1Þ ;

ð5:2Þ

and
where M ð1Þ is the one-skeleton of M and X ð1Þ the one-skeleton of X : We will begin
by analyzing these conditions and their implications. We ﬁrst note that the analogues
for M of the conditions (5.1) and (5.2), namely#M G oNð5:3ÞdimðM ð1Þ Þp2;

ð5:4Þ

and

have a very simple and elegant interpretation in terms of the isotropy representationsof G at ﬁxed points of M:
Theorem 5.1. The conditions #M G oN and dimðM ð1Þ Þp2 are satisfied if and only if,for every pAM G ; the weights a* i;p ; i ¼ 1; y; d of the isotropy representation of G on
Tp M are pair-wise linearly independent, that is for iaj; a* i;p is not a multiple of a* j;p :For the proof of this, see [GZ]. When M satisﬁes the two conditions (5.1) and(5.2), we say that M is a GKM manifold. Let ZG be the weight lattice of G: By themod 2 reduction of a weight a* AZG ; we mean its image a in ZG =2ZG : We will prove areal analogue of Theorem 5.1.
Theorem 5.2. Suppose M satisfies the hypotheses of Theorem 5.1. Then the conditions
X GR ¼ M G and X ð1Þ ¼ X -M ð1Þ are satisfied if and only if, for every pAM G ; themod 2 reduced weights, ai;p ; are all distinct and non-zero.
Proof. Let Y be a connected component of M GR : Then Y is a G-invariant symplectic
submanifold of M; and the action of G on it is Hamiltonian, so it contains at least
one G-ﬁxed point p: However, the hypotheses above imply that the linear isotropy

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action of GR on Tp M has no ﬁxed points other than the origin. Hence, dimðY Þ ¼ 0
and Y ¼ fpg: This argument applies to all the connected components of M GR ; hence
the connected components are just the ﬁxed points of G; and thus X GR ¼ M G :The proof that X ð1Þ ¼ X -M ð1Þ is similar. Let HR be a subgroup of GR of index 2;
and let Y be a connected component of M HR : Then Y is a G-invariant submanifoldof M; and because s3tg ¼ tg1 3s; it is also s-invariant. Let pAY be a G-ﬁxed point,
and letTp M ¼ V1 "?"Vdbe the decomposition of Tp M into the two-dimensional weight spaces corresponding
to the a* i;p : By the hypotheses on the reduced weights ai;p ; eitherðTp MÞHR ¼ f0g;
in which case Y ¼ fpg orðTp MÞHR ¼ Vi ¼ Tp Y

ð5:5Þfor some i: Let wi be the character of G associated with the representation of G on Vi
and let H ¼ kerðwi Þ: Then HR CH andðTp MÞH ¼ Vi :Thus, by (5.5), Y is the connected component of M H containing p; and in particular,
Y is contained in M ð1Þ : Thus,
Y s DX -M ð1Þ :Applying this argument to all index 2 subgroups HR of GR and all connectedcomponents of the ﬁxed point sets of these groups, one obtains the inclusion
X ð1Þ DX -M ð1Þ :The reverse inclusion is obvious. This completes the proof.

&

The hypotheses of Theorem 5.2 impose some rather severe restrictions on the
manifold M: For instance, the cardinality of the set of mod 2 reduced weights,
ZG =2ZG ; is 2n : Therefore, since the reduced weights ai;p are distinct and non-zero fori ¼ 1; y; d; we must have that dr2n  1: Hence,dimðMÞ ¼ 2dp2nþ1  2:

ð5:6ÞFor example, if n ¼ 2; then dimðMÞp6: This leads us to make the following
deﬁnition.

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Deﬁnition 5.3. If M is a GKM manifold, and if for every pAM G ; the mod 2 reduced
weights, ai;p ; are all distinct and non-zero, we will say that M is a mod 2 GKM
manifold.
Next, we show that relatively few compact homogeneous symplectic manifolds (e.g.
coadjoint orbits) are mod 2 GKM manifolds. Consider coadjoint orbits of theclassical compact simple Lie groups associated with the Dynkin diagrams An ; Bn ; Cn
and Dn : Let ei ; for i ¼ 1; y; n; be the standard basis vectors of Rn : The positive roots
associated to the Dynkin diagram An consist ofei  ej ;ioj;

so their mod 2 reductions are distinct and non-zero. However, for Bn ; Cn ; and Dn ;this list of positive roots contains
ei  ejandei þ ej ; ioj;

so we conclude
Theorem 5.4. Each coadjoint orbit of SUðnÞ is a mod 2 GKM space. However, forother compact simple Lie groups, no maximal coadjoint orbit can be a mod 2 GKM
space.
On the other hand, on a more positive note, one has
Theorem 5.5. If M is a non-singular projective toric variety, then it is a mod 2 GKM
space.
Proof. If M is a non-singular toric variety, the weights a* i;p ; i ¼ 1; y; n; are a Z-basis
for ZG ; so their images in ZG =2ZG are a Z=2Z basis of ZG =2ZG : &This theorem, combined with Theorem C, gives us a new descriptionof the equivariant cohomology of a real toric variety. The ordinary andGR -equivariant cohomology of these real loci has been computed by Davisand Januszkiewicz [DJ]. Their description of these rings is analogous to
Danilov’s description of the ordinary and G-equivariant cohomology of the original
toric varieties [Da].
We will now prove a real locus version of the GKM theorem with Z=2Z
coefﬁcients. Recall from Section 1 that Theorem 1.1 of GKM characterizes the imageof i : HG ðM; CÞ-HG ðM G ; CÞ in terms of the weights of the isotropy representationsof G on the tangent spaces at the ﬁxed points.
To prove an analogue of this for the real locus of a symplectic manifold, wemust ﬁrst compute the Z=2Z-equivariant cohomology with Z=2Z coefﬁcientsof RP1 : Recall that S1 acts on CP1 by y ½z0 : z1 ¼ ½z0 : eiy z1 : This is a Hamiltonian
action, with respect to the Fubini-Study symplectic form on CP1 : Furthermore,ARTICLE IN PRESS
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389

complex conjugation is an anti-symplectic involution on CP1 ; with ﬁxed point set
RP1 : There is a residual action of Z=2Z on RP1 DS 1 which reﬂects S 1 about
the y-axis.
Lemma 5.6. Let N and S denote the fixed points of the Z=2Z action on RP1 : Then theimage of the mapi : HZ=2Z
ðRP1 ; Z=2ZÞ-HZ=2Z
ðN; Z=2ZÞ"HZ=2Z
ðS; Z=2ZÞ

is the set of pairs ð fN ; fS Þ such thatfN þ fS Ax Z=2Z½x :Proof. It is clear that the constant functions are equivariant classes in0
HZ=2ZðRP1 ; Z=2ZÞ:0Furthermore, we know that dim HZ=2Z
ðRP1 ; Z=2ZÞ ¼ 1; and so these are the onlyiequivariant classes. Finally, dim HZ=2Z
ðRP1 ; Z=2ZÞ ¼ 2 for i40; and so indeed, thecondition stated is the only condition of pairs ð fN ; fS ÞAHZ=2Z
ðN; Z=2ZÞ"
HZ=2Z ðS; Z=2ZÞ: &Theorem C identiﬁes the image of the mapi : HG R ðX ; Z=2ZÞ-HG R ðX GR ; Z=2ZÞ
in terms of weights of isotropy representations of GR on the tangent spaces at theﬁxed points.
Proof of Theorem C. The result follows immediately from Corollary 4.8 andLemma 5.6. &The results of this section and the previous section have been proved
independently by Schmid [S]. Schmid uses an equivariant Morse theoretic approach,and consequently the proofs are quite different.
As a result of equivariant formality, we get two corollaries of Theorem C
concerning the relation between the ring structure of the cohomology of M and thecohomology of X :Corollary 5.7. Suppose that M is a GKM manifold and a mod 2 GKM manifold. Then
there is a graded ring isomorphism
HG2 ðM; Z=2ZÞDHG R ðX ; Z=2ZÞ:ARTICLE IN PRESS390

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Corollary 5.8. Suppose that M is a GKM manifold and a mod 2 GKM manifold. Then
there is a graded ring isomorphism
H 2 ðM; Z=2ZÞDH  ðX ; Z=2ZÞ:Note that this last corollary strengthens Duistermaat’s original result from an
isomorphism of vector spaces to an isomorphism of rings.
Remark 5.9. Some of the results of this section, most importantly Theorem C, arevalid not only for the real locus X of a Hamiltonian G-manifold, but more generally
for any compact GR -manifold X which satisﬁes the following properties:(1) X is equivariantly formal;(2) X GR is ﬁnite; and(3) the weights of X satisfy the properties of a mod 2 GKM manifold.
In this situation, we may still characterize the structure of the one-skeleton.
Theorem C still follows from injectivity and the Chang–Skjelbred theorem.

6. Real GH
Goldin and Holm generalize Theorem 1.1 to the case where the one-skeleton has
dimension at most 4: The goal of this section is to prove a real version of the Goldin–
Holm theorem with Z=2Z coefﬁcients. Again, we require the hypotheses that theðZ=2ZÞn -ﬁxed points of the real locus are the same as the G-ﬁxed points of M as in(5.1); and that the real locus of the one-skeleton is the same as the one-skeleton of thereal locus, as in (5.2). Finally, we require#M G oN
anddimðM ð1Þ Þp4:

If a manifold satisﬁes these last two hypotheses, we will say that it is a GH manifold.
These hypotheses have a nice interpretation in terms of the isotropy representationsof G at the ﬁxed points of M:
Theorem 6.1. The conditions #M G oN and dimðM ð1Þ Þp4 are satisfied if and only if
the weights ai;p of the isotropy representation of G on Tp M have the property that every
three span a vector subspace of dimension at least two.
These hypotheses on M have real analogues, namely that #X oN and the oneskeleton X ð1Þ of X is at most two-dimensional. We will state without proof thefollowing real analogue of Theorem 6.1.

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Theorem 6.2. Suppose that M satisfies the hypotheses of Theorem 6.1. If theconditions X GR ¼ M G and X ð1Þ ¼ M ð1Þ -X are satisfied, then for every pAM G ; the
mod 2 reduced weights a#
i;p are all non-zero, and each element of SððGR Þ Þ ¼
Z=2Z½x1 ; y; xn appears no more than twice.
The proof of this theorem is nearly identical to that of Theorem 5.2. The
hypotheses of this theorem, although weaker than those of Theorem 5.2, still impose
restrictions on the manifold M: The cardinality of the set of mod 2 reduced weights is
2n : Since the weights are non-zero, and each weight can appear at most twice,dp2 ð2n  1Þ;and sodimðMÞ ¼ 2dp2 ð2 ð2n  1ÞÞ ¼ 2nþ2  4:
For instance, if n ¼ 2; dimðMÞp12: We will now show an example where thecondition that the reduced weights be non-zero is not satisﬁed.
Example. Consider CP2 with homogeneous coordinates ½z0 : z1 : z2 : Let T ¼ S 1 act
on CP2 byeiy ½z0 : z1 : z2 ¼ ½eiy z0 : z1 : eiy z2 :This action has three ﬁxed points: ½1 : 0 : 0 ; ½0 : 1 : 0 and ½0 : 0 : 1 :The weights at these ﬁxed points are as follows:
Fixed point

Weights

p1 ¼ ½1 : 0 : 0x; 2xp2 ¼ ½0 : 1 : 0

x; x

p3 ¼ ½0 : 0 : 1

2x; xwhere we have identiﬁed t with degree one polynomials in C½x : As cohomology
elements, these are assigned degree two. Using Theorem 6.5 below, we can compute
the S 1 equivariant cohomology of CP2 as follows. The image of the equivariant
cohomology HS1 ðCP2 Þ inHS1 ðf p1 ; p2 ; p3 gÞD

3
M

C½x

i¼1

is the subalgebra generated by the triples of functions ð f1 ; f2 ; f3 Þ such thatfi  fj Ax C½x for every i and j
and
f1
f2
f3
 2 þ 2 AC½x :
2
2x
x
2x

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However, when we try to compute the Z=2Z equivariant cohomology of RP2 ; thereal locus of CP2 ; we run into a problem. The mod 2 reduced weights are as follows:
Fixed point

Weights

p1 ¼ ½1 : 0 : 0x; 0p2 ¼ ½0 : 1 : 0x; x

p3 ¼ ½0 : 0 : 10; xThe problem with this Z=2Z action on RP2 is that it no longer has isolated ﬁxed
points. There is an entire RP1 which is ﬁxed by this Z=2Z action. Thus, we cannot
compute the Z=2Z equivariant cohomology of RP2 using these methods.
We make the following deﬁnition, analogous to the deﬁnition of mod 2 GKM
manifolds given in Section 5.
Deﬁnition 6.3. Suppose that M is a GH manifold, and furthermore that X GR ¼ M G
and X ð1Þ ¼ M ð1Þ -X : In this case, we will say that M is a mod 2 GH space.
Recall the following properties about the G-equivariant cohomology of manifolds
with one-skeleta of dimension at most 4: These are proved in [GH], although thereader is cautioned to the different notation used there. First, we compute the S 1
equivariant cohomology of a 4-manifold, and then we use this computation todetermine the equivariant cohomology of any manifold with one skeleton ofdimension at most 4.
Lemma 6.4. Let X be a compact, connected symplectic 4-manifold with an effective
1
Hamiltonian S 1 action with isolated fixed points X S ¼ fp1 ; y; pd g: The map
1i : HS1 ðX Þ-HS1 ðX S Þ induced by inclusion is an injection with image
(


)

d
X
fi

ð f1 ; y; fd ÞASðs Þ fi  fj Ax C½x ;ASðs Þ ;

ai ai
i¼1
i¼1 1 2dM 

where ai1 and ai2 are the (linearly dependent) weights of the S ¼ S1 isotropy action on
Tpi X :
Theorem 6.5. Let M be a compact, connected symplectic manifold with an effective
Hamiltonian G-action. Suppose further that the G-action has only isolated fixed
points M G ¼ f p1 ; y; pd g and that the one skeleton has dimension at most 4:
Let fi AHG denote the restriction of f AHG ðMÞ to the fixed point pi : The imageof the injection i : HG ðMÞ-HG ðM G Þ is the subalgebra of functions

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393

ð f1 ; y; fd ÞA"di¼1 Sðg Þ which satisfy
8 >< pH ð fij Þ ¼ pH ð fik Þ
l
P
fij
>
ASðg Þ:ij ij
j¼1 a1 a2G
if f pi1 ; y; pic g ¼ ZH;
Gif f pi1 ; y; pic g ¼ ZH
and dim ZH ¼ 4iifor all HCG codimension-1 tori and all connected components ZH of M H ; where a1j ; a2j
are the (linearly dependent) weights of the G action on Tpij ZH ; and pH : h+g isinclusion.
We can use these computations to compute the ðZ=2ZÞn equivariant cohomology
of a mod 2 GH manifold.
Lemma 6.6. Let M be a compact, connected symplectic 4-manifold with an effective
1
Hamiltonian S 1 action with isolated fixed points M S ¼ fp1 ; y; pd g: Suppose further
that M is a mod 2 GH manifold with real locus X : The mapi : HZ=2Z
ðX ; Z=2ZÞ-HZ=2Z
ðX Z=2Z ; Z=2ZÞ

induced by inclusion is an injection with image
9
 fi  fj Ax Z=2Z½x ; >

=
d
M
 d
;
ð f1 ; y; fd ÞA
Z=2Z½x  P
fi

>
>
:
i¼1
 i¼1 ai ai AZ=2Z½x ; ;
1 2
8
>
<

ð6:1Þ

where ai1 and ai2 are the linearly dependent weights of the Z=2Z isotropy representation
on Tpi X : (In this case, ai1 ¼ ai2 ¼ x:)
Proof. The map i is injective because X is equivariantly formal. We know that the fimust satisfy the ﬁrst condition because the functions constant on all the vertices are0ðX ; Z=2ZÞ ¼ 1: The second
the only equivariant classes in degree 0; as dim HZ=2Z
condition is necessary as a direct result of the Z=2Z version of the localization
theorem proved in Section 4. Note that this condition gives us one relation in degree 1
cohomology. A dimension count shows us that these conditions are sufﬁcient. As anSððZ=2ZÞ Þ-module, HZ=2Z
ðX ; Z=2ZÞDH  ðX ; Z=2ZÞ#HZ=2Z
ð pt; Z=2ZÞ: Thus, theequivariant Poincaré polynomial is
Z=2Z

Pt

ðX Þ ¼ ð1 þ ðd  2Þt þ t2 Þ ð1 þ t þ t2 þ ?Þ
¼ 1 þ ðd  1Þt þ dt2 þ ? þ dtn þ ?:

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As HZ=2Z
ðX ; Z=2ZÞ is generated in degree 1, the d  1 degree 1 classes given by theð f1 ; y; fd Þ subject to the localization condition generate the entire cohomology ring.
Thus, we have found all the conditions. &We now prove Theorem D, computing the cohomology of any mod 2 GH
manifold. We will show that the image of i : HG R ðX Þ-HG R ðX GR Þ is the subalgebraof functions ð f1 ; y; fd ÞA"di¼1 SðGR Þ which satisfy
8 
GR

>
< pHR ð fij Þ ¼ pHR ð fik Þ if f pi1 ; y; pil g ¼ ZHR ;
l
fij
P
GR

>
if f pi1 ; y; pil g ¼ ZH
and dim ZHR ¼ 2:
ij ij ASðGR ÞR
j¼1 a1 a2
for all subgroups HR of GR of order jHR j ¼ 2n1 and all connected components ZHRof X HR :Proof of Theorem D. This follows immediately from Corollary 4.8 and Lemma
6.6. &
There are two immediate corollaries in this setting, analogous to Corollaries 5.7
and 5.8.
Corollary 6.7. Suppose that M is a GH manifold, and that M G ¼ X GR and M ð1Þ -X ¼
X ð1Þ : Then there is a graded ring isomorphism
HG2 ðM; Z=2ZÞDHG R ðX ; Z=2ZÞ:Corollary 6.8. Suppose that M is a GH manifold, and that M G ¼ X GR and M ð1Þ -X ¼
X ð1Þ : Then there is a graded ring isomorphism
H 2 ðM; Z=2ZÞDH  ðX ; Z=2ZÞ:7. Applications to string theory
Consider the Z=2Z action on T n ; which reﬂects each copy of S1 : Then theequivariant cohomology ring
HZ=2ZðT n ; Z=2ZÞ

classiﬁes all possible orientifold conﬁgurations of Type II string theories,
compactiﬁed on T n : See Section 3 and Appendix C of [dB] for more details.

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Fig. 1. This shows the GKM graph and the weights for ðCP1 Þ3 :Yang-Hui He pointed out this example to us. Using the results of Section 5, we can
now compute this equivariant cohomology.
First, we recognize T n as the real locus of M ¼ CP1  ?  CP1 ¼ ðCP1 Þn : This
space M has a natural T n action, where the ith copy of S 1 acts in the standard
fashion on the ith copy of CP1 : We can compute the ðZ=2ZÞn -equivariant
cohomology of this space quite easily. The GKM graph associated to ðCP1 Þn with
the T n action described above is the n-dimensional hypercube. The vertices
correspond to the binary words of length n: Two binary words are connected byan edge if they differ in exactly one bit. Suppose v and w differ in exactly the ith bit.
Then the weight associated to the edge ðv; wÞ is xi : Thus, when n ¼ 3; the GKM
graph and weights are shown in the ﬁgure above (Fig. 1).
Note that the reduced weights are all non-zero and are distinct in ZG =2ZG : Thus,
we can apply Theorem C to compute
HðZ=2ZÞn ðT n ; Z=2ZÞ:That is, the equivariant cohomology is the set of functions f : V -Z=2Z½x1 ; y; xn
such that for every edge ðv; wÞAE; we havef ðvÞ þ f ðwÞAxi Z=2Z½x1 ; y; xn :We can now consider the copy of Z=2Z sitting diagonally inside ðZ=2ZÞn :This copy of Z=2Z acts on T n ; and this is the action that originally interested
physicists. We can now compute the Z=2Z-equivariant cohomology simply by
projecting
p : SðððZ=2ZÞn Þ Þ ¼ Z=2Z½x1 ; y; xn -Z=2Z½x ¼ SððZ=2ZÞ Þ
where xi gets sent to x: Then
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D. Biss et al. / Advances in Mathematics 185 (2004) 370–399

8. Recreational applications
We will describe below a (somewhat idealized) real world application of the resultsof Section 5. Let S ¼ f1; y; ng be a list of companies whose stocks are being tradedin the stock market. Let Ai for i ¼ 1; y; d be investors, and let Si DS be the portfolioof Ai : Suppose that for certain pairs of investors Ai and Aj ; the symmetric differenceðSi  Sj Þ,ðSj  Si Þ ¼ Se ;e ¼ ði; jÞ;ð8:1Þ

i.e. the relative status of the portfolios of Ai and Aj ; is given. To what extent does thisinformation determine the portfolios Si ? It is easy to see that it cannot uniquelydetermine the Si ’s. If S1 ; y; Sn is one solution to (8.1), one gets another solution bytaking a ﬁxed subset S0 of S and replacing the Si ’s by the symmetric difference
Si0 ¼ ðSi  S0 Þ,ðS0  Si Þ:

ð8:2Þ

Therefore, we will slightly rephrase this question. Let E be a collection of twoelement subsets of S; and suppose that for every eAE; one is given a subset Se of S:
List all solutions S1 ; y; Sn of the Boolean identities (8.1). We will say two solutions
are identical if they satisfy (8.2). Note, by the way, that (8.2) can be rewritten as
Si ¼ ððSi0  S0 Þ,S0 Þ  Si0 :Hence, we have deﬁned an equivalence relation.
One can inject an element of randomness into this problem by positing that, fore ¼ ði; jÞAE;ðSi  Sj Þ,ðSj  Si ÞAfSe ; |g:ð8:3Þ

In other words, either the symmetric difference is given by (8.1) or Si ¼ Sj : Again,
the problem is to list all possibilities for the Si ’s. Clearly, the solutions of (8.3)
contain the solutions to (8.1); so by solving (8.3), one gets an upper bound on thenumber of solutions to (8.1). Moreover, there are a lot of trivial solutions of (8.3),
namely,Si ¼ S0 ;i ¼ 1; y; n;ð8:4Þ

where S0 is, as above, a ﬁxed subset of S: These can immediately be discarded as
potential solutions of (8.1).
There is an elegant way of reformulating (8.1) and (8.3) in the language of mod 2
arithmetic. Let G be the graph whose vertices are the Ai ’s and whose edges are themembers of E: For every edge eAE; let ae be the element of Z=2Zn whose kth
coordinate is 1 if and only if kASe and ai the element of Z=2Zn whose kth coordinate
is 1 if and only if kASi : Then (8.1) is equivalent toai þ aj ¼ ae

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397

and (8.3) is equivalent toai þ aj ¼ lae ;lAZ=2Z:

ð8:6Þ

In particular, (8.3) becomes an identity of the type described in Theorem C, themultiple of ae on the right being in the degree zero component of Z=2Z½x1 ; y; xn :Now let M be a GKM manifold acted on by an n-torus T; and let G be its
associated graph. If the ae ’s in (8.6) are the weights of GR which were deﬁned in
Theorem C the results of Section 5 tell us that solutions of (8.6) can be identiﬁed withelements of
HG1 R ðM s ; Z=2ZÞ:We will use this observation to determine the solutions of (8.6) in a couple simple,
but interesting, examples.
Example (The complete graph on n vertices). Consider M ¼ CPn1 as a T n
manifold (ignoring the fact that the diagonal subgroup of T n acts trivially). The
corresponding graph is the complete graph on n vertices: its vertices are A1 ; y; Anand every pair of vertices is joined by an edge. The weights aði;jÞ from Theorem C are
just xi þ xj ; so conditions (8.1) becomeðSi  Sj Þ,ðSj  Si Þ ¼ fi; jg:

ð8:7Þ

The solutions of (8.7) are in one-to-one correspondence with the elements of
HG1 R ðRPn1 ; Z=2ZÞ
or, alternatively, ofðH 0 ðRPn1 ; Z=2ZÞ#Z=2Zn Þ"H 1 ðRPn1 ; Z=2ZÞ:ð8:8Þ

The elements of the ﬁrst summand correspond to the trivial solutions of (8.7). So if
we identify solutions which are equivalent in the sense of (8.2), the non-trivial
solutions of (8.7) correspond to non-zero elements of H 1 ðRPn1 ; Z=2ZÞ: However,H 1 ðRPn1 ; Z=2ZÞ ¼ Z=2Z; so there is just one non-trivial solution up to equivalence,and it is given by
Si ¼ fig;
this is also the unique solution of (8.1) up to equivalence.
Example (The permutahedron). Let M be the complex ﬂag variety UðnÞ=T n ; andconsider the T n action on M by left multiplication. The graph associated with M is
the permutahedron. Its vertices are the elements of the symmetric group, Sn ; and two
vertices s and t are joined by and edge if ts1 is a transposition. If e is the edge
joining s to t and ts1 is the transposition switching i and j; then as in the previous

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example, ae ¼ xi þ xj ; so condition (8.1) becomesðSs  St Þ,ðSt  Ss Þ ¼ fi; jg:
As in the previous example, the non-trivial solutions to (8.3) can be identiﬁed with
the non-zero elements of H 1 ðM s ; Z=2ZÞ; and since M s is the real ﬂag variety,H 1 ðM s ; Z=2ZÞDZ=2Zn1 :If one thinks of Z=2Zn1 as the quotient of Z=2Zn by the diagonal subgroup
ðl; y; lÞ; with lAZ=2Z; the solutions corresponding to aAZ=2Zn modðl; y; lÞ isgiven bySs ¼ sðS0 Þ;

ð8:9Þ

where kAS0 if and only if the kth coordinate of a is 1: Note, by the way, if we replace
a by a þ ð1; y; 1Þ; (8.9) becomesSsc ¼ sðS0c Þ;ð8:10Þ

where Ssc and S0c are the complements of Ss and S0 in S; so (8.9) and (8.10) areequivalent. It is easy to see by inspection that none of the solutions (8.9) of (8.3) arealso solutions of (8.1).
An interesting special case of this example is the complete bipartite graph K3;3 : In
this case, the vertices of G are A1 ; A2 ; A3 and B1 ; B2 ; and B3 ; the edges are all pairs
ðAi ; Bj Þ; and the sets Se are(fi; jg;iaj;
SAi ;Bj ¼
f1; 2; 3g  fig; i ¼ j:
In this example, (8.3) has one non-trivial solution, up to equivalence, namely SAi ¼
SBi ¼ fig:
Acknowledgments
The authors thank the anonymous referees for useful comments and suggestions.

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