BORDISM OF SEMI-FREE S1-ACTIONS
DEV SINHA
Abstract.We calculate the geometric and homotopical (or stable) bordism r*
*ings associated to semi-
free S1 actions on complex manifolds, giving explicit generators for the *
*geometric theory. To calculate
the geometric theory, we prove a case of the geometric realization conjec*
*ture, which in general would
determine the geometric theory in terms of the homotopical. The determina*
*tion of semi-free actions with
isolated fixed points up to cobordism complements similar results from sy*
*mplectic geometry.
1.Introduction
In this paper we describe both the geometric and homotopical bordism rings as*
*sociated to S1-actions
in which only the two simplest orbit types, namely fixed points and free orbits*
*, are allowed. Our work
is of further interest in two different ways. To make the computation of geomet*
*ric semi-free bordism, in
Theorem 3.11 we prove the semi-free case of what we call the geometric realizat*
*ion conjecture (Conjec-
ture 2.7), which if true in general would determine the ring structure of geome*
*tric S1-bordism from the
ring structure of homotopical S1-bordism given in [11]. Additionally, an applic*
*ation of our results to semi-
free actions with isolated fixed points which we state now gives results parall*
*el to results from symplectic
geometry [10]. Let P(C æ) denote the space of complex lines in C æ where æ *
*is the standard complex
representation of S1 (in other words, the Riemann sphere with S1 action given b*
*y the action of the unit
complex numbers.)
Theorem 1.1. Let S1 act semi-freely with isolated fixed points on M, compatible*
* with a stable complex
structure on M. Then M is equivariantly cobordant to a disjoint union of produc*
*ts of P(C æ).
This result should be compared with the second main result of [10], which sta*
*tes that when M is con-
nected a semi-free Hamiltonian S1 action on M implies that M has the same Borel*
* equivariant cohomology
and equivariant Chern classes as a product of such P1's. Based on their results*
* and ours, we make the
following.
Conjecture 1.2. A semi-free Hamiltonian S1 action with isolated fixed points on*
* a connected manifold is
equivariantly diffeomorphic to a product of P(C æ)'s.
As Theorem 1.1 lead us to the more general computation of Theorem 3.12, it wo*
*uld also be interesting
to see if there is an analog of Theorem 3.12 for Hamiltonian S1-actions. In ge*
*neral, the symplectic
and cobordism approaches to transformation groups have remarkable overlaps in l*
*anguage (for example,
localization by inverting Euler classes of representations plays a key role in *
*each theory), though the same
words sometimes have different precise meanings. A synthesis of these technique*
*s could perhaps address
Conjecture 1.2 or other interesting questions within transformation groups.
The author would like to thank Jonathan Weitsman for stimulating conversation*
*s.
2. Review of complex equivariant bordism
The foundational results of this section are taken from [7], and the computat*
*ional results are taken from
[11]. If V is a representation of G equipped with a G-invariant inner product, *
*let SV denote its one-point
1
2 DEV SINHA
compactification, let D(V ) be the unit disk in V , and let S(V ) be the bounda*
*ry of D(V ), namely the unit
sphere in V .
Let U,G*denote geometric complex equivariant bordism (see for example chapte*
*r 28 of [7]), the bordism
ring whose representatives are stably complex G-manifolds up to cobordisms in w*
*hich the bounding man-
ifolds have stably complex G-actions which extend the actions on their boundari*
*es. By equipping these
manifolds with maps to a space X we define an equivariant homology theory U,G**
*(X), which extends to a
homology theory on pairs by using manifolds with boundary. Note that this equiv*
*ariant homology theory
has suspension isomorphisms only when G acts trivially on the suspension coordi*
*nate.
Let MUG*denote stable complex equivariant bordism, represented by a direct li*
*mit of homotopy classes
of maps from spheres SV to Thom spaces (see for example section 3 of [11]). Alt*
*ernately, by a result of
Bröcker and Hook [2], MUG*(X) is isomorphic to a direct limit which has non-tri*
*vial G-suspensions built
in, namely limV U,G*(X x D(V ), X x S(V )), where V ranges over the real repres*
*entations of G. Thus,
MUG*is also represented by stably complex G-manifolds M with boundary equipped *
*with a map to D(V )
for some representation V such that the boundary of M maps to S(V ). Through ei*
*ther the Pontryagin-
Thom construction or through the Bröcker-Hook result, there is a canonical a ma*
*p OE: U,G*! MUG*. The
following result allows us to use homotopy theory to effectively study G-manifo*
*lds.
Theorem 2.1 (Comeza~na, Theorem 28.5.4 of [7]). If A is abelian, the map OE: U*
*,A*! MUA*is injective.
What makes MUA*at all manageable is that it in turn maps injectively to a loc*
*alization which is
computable. To set this stage, we recall some of the main results from [11].
Euler classes play a central role in equivariant bordism. Vector spaces with *
*G-action, viewed as vector
bundles over a point, may have interesting characteristic classes in the equiva*
*riant cohomology of a point.
For a complex representation V let eV 2 MUnG(pt.) = MUG-n, where n is the dimen*
*sion of V over the
reals, denote the Euler class of V . Under the Bröcker-Hook isomorphism, eV is *
*represented by a point (as
a zero-manifold) mapping to zero in D(V ). It is intuitively clear, and shown i*
*n chapter 15 of [7], that eV
is non-trivial if V G= 0. Thus, MUG*contains many classes in arbitrarily negati*
*ve degrees. Euler classes
multiply by the rule eVeW = eV W. Let S be the multiplicative set of non-trivi*
*al Euler classes. Let Zn,V
be the class in MUG*represented by P(Cn V ), the space of lines in Cn V wit*
*h induced G-action.
Theorem 2.2 (Theorem 1.2.5 of [11]). There are inclusions of MUT*-algebras.
MU*[eV, Zn,V] ! MUT*~!S-1MUT*~=MU*[eV1, Zn,V],
where V ranges over non-trivial irreducible representations of T, n 2, and ~ *
*is the canonical localization
map.
Thus, understanding of MUT*follows from understanding of divisibility by Eule*
*r classes. Crucial insight
is provided by the following theorem, which is proved by applying MU*Tto the co*
*fiber sequence S(V ) !
* ! SV . Let K(V ) denote the kernel of the representation V .
Theorem 2.3 (Theorem 1.2 of [11]). There is an isomorphism of MUT*=(eV) with MU*
*K(V*), where the
quotient map from MUT*to MUT*=(eV) coincides with the restriction map rV from M*
*UT*to MUK(V*).
By carefully splitting the restriction map rV and composing that splitting wi*
*th rV itself, in [11] we
define idempotent operations fiV :MUT*! MUT*such that x - fiV(x) is divisible b*
*y eV. By Theorem 2.2,
the quotient of x - fiV(x) by eV is unique, and we call that quotient V(x). We*
* say V is a Conner-Floyd
operation. If I = V1, . .,.Vk is a finite sequence of non-trivial irreducible *
*representations let I(x) =
Vk Vk-1. . .V1x. Fix an ordering on the non-trivial irreducible representation*
*s of T which includes the
relation V < W if K(V ) K(W). Call a finite sequence of representations admis*
*sible if it respects this
ordering.
The following is the main theorem of [11].
BORDISM OF SEMI-FREE S1-ACTIONS *
* 3
Theorem 2.4. MUT*is generated as an MU*-algebra by the classes I(eV) and I(Zn*
*,V), where V
ranges over non-trivial irreducible representations, I ranges over all admissis*
*ible sequences of non-trivial
irreducible representations, and n ranges over natural numbers.
Relations include the following:
(1)eV V(x) = x - fiV(x)
(2) V(fiV(x)) = 0
(3) V(eV) = 1
(4) V(x)y = (x - fiV(x)) V(y) + V(x)fiV(y)
(5) V Wx = WV x - WV fiW (x) - WV (eW fiV( W x))
where V and W range over non-trivial irreducible representations of T and x and*
* y are any classes in
MUT*. For T = S1, these relations are complete.
1
Note that while MUS1*is known as a ring, U,S*is not known because it is1not *
*known which classes
I(x), and in particular which I(eV) can be realized by classes in U,S*. As w*
*ith much of the theory,
this question is best understood by applying1the localization map ~. By Proposi*
*tions 4.13 and 4.5 of [11],
one may compute the image of a class in U,S* MUS1*under ~ by investigating fi*
*xed sets and normal
bundle data of that manifold. Indeed, this is one of the main results of the pa*
*per [4].
Definition 2.5. Let P* denote the sub-algebra MU*[e-1V, Zn,V] of MU*[eV1, Zn,V].
In other words P* is the sub-algebra of S-1MUT*generated by those generators *
*which are in positive
degrees. Note that this algebra is called F* in [11].
1
Proposition 2.6 (See Proposition 4.13 of [11]). The image of U,S*under ~ lies *
*in P*.
We make the following1conjecture, a converse to Proposition 2.6, which would *
*determine the ring
structure of U,S*from that of MUS1*and its image under ~.
1
Conjecture 2.7 (The realization conjecture). If ~(x) 2 P* then x 2 U,S*.
As evidence for this conjecture, we will prove it in the semi-free setting.
3.Semi-free bordism
We now focus on the case where T = S1 and the S1 action on manifolds in quest*
*ion is semi-free, so
that points are either fixed or freely acted upon by S1.
1
Definition 3.1. Let SF*denote the subring of U,S*of bordism classes represent*
*ed by semi-free actions.
For a semi-free action, the normal bundle of a fixed set will have representa*
*tion type on the fiber of æ
or æ*, where æ is the standard one-dimensional representation of S1 and æ*is it*
*s conjugate.
Note here that bordisms between1our semi-free manifolds are allowed to have g*
*eneral S1-action, so we
are looking at the image in U,S of the theory which we may call SF!*in which *
*all manifolds in question
have semi-free action. But in fact, by looking at the families exact sequence f*
*or the family consisting of
all of S1 and the identity subgroup (see chapter 15 of [7], or [12]) both of th*
*ese theories fit in long exact
sequences
0 _ ! 21
G @
. .!.MU*-1(BS1) i! SF*~!MU*@ BU(n) A ! MU*-2(BS1) ! . .,.
n>0
which map to each other and thus can be shown to be isomorphisms by the five-le*
*mma. Here, if we have M
mapping to BS1 we may pull back the canonical S1-bundle to get a free S1 manifo*
*ld, so that i is inclusion
of free S1-manifolds into the semi-free theory. And ~ in this setting sends a s*
*emi-free bordism class to the
4 DEV SINHA
bordism class of the normal bundle (split according to appearance of æ and æ*in*
* the fiber). Finally, @
sends a manifold with a direct sum two bundles over it (classified by a map som*
*e BU(i) x BU(j)) to the
free S1 manifold given by the sphere bundle of that bundle, where S1 acts as æ *
*on the first factor and æ*
on the second. i F j 1
In fact, by identifiying MU* n>0BU(n) 2 as a sub-ring of P* S-1MUS* (aga*
*in, see Proposi-
tion 4.13 of [11]) we have the following.
Lemma 3.2. The image under ~ of SF*lies in the sub-ring PSF*= MU*[e-1æ, e-1æ*,*
* Zn,æ, Zn,æ*] of P*.
We are lead to the following.
Definition 3.3. Let MUSF*be the subring of MUS1*which maps under ~ to Z[eæ1, eæ*
*1*, Zn,æ, Zn,æ*].
Our main results are computations of MUSF*and then, remarkably, SF*as MU*-al*
*gebras. First, we
pause to consider semi-free manifolds with isolated fixed points. In fact, at f*
*irst we analyzed this case
because of its independent interest [10] and then realized it could be used as *
*a base case in a filtration to
compute SF*. Later, we found that we could compute Sf*more directly from Theo*
*rem 2.4 and deduce
Theorem 1.1 from that computation. Now we choose to present the isolated fixed *
*point case independently
from the more general semi-free case (Theorem 3.12, as the two approaches are c*
*omplementary.
Under the identification of Lemma 3.2, semi-free actions with isolated fixed *
*points have image under ~
which sit in the subring Z[e-1æ, e-1æ*]. In particular ~(P(C æ) = e-1æ+ e-1æ*.
Theorem 3.4. The intersection of ~(MUS1*) with the subring Z[e-1æ, e-1æ*] is th*
*e subring Z[e-1æ+ e-1æ*].
This theorem, along with Theorem 2.1 and Theorem 2.2, implies Theorem 1.1 and*
* thus characterizes
semi-free actions with isolated fixed points up to cobordism. Our main tool in *
*this direct proof is application
of Theorem 2.3, which for V = æ or æ*says that reduction modulo eæor eæ*coincid*
*es with the augmentation
map from MUS1*to MU*.
Proof of Theorem 3.4.Let R denote the subring Z[e-1æ, e-1æ*] of S-1MUS1*, and l*
*et Q denote the subring
Z[e-1æ+ e-1æ*] of R, so Q = ~(Z[P(1 æ)]).
Since R is graded and lies in positive degrees, we may induct by degree, focu*
*sing on homogeneous
elements. Suppose that a0e-næ+ a1e-(n-1)æeæ*+ . .+.ane-næ*is equal to ~(x). C*
*onsider y = eæ*(x -
a0[P(C æ)]n). The image ~(y) is in R and is in degree 2(n - 1), thus by induc*
*tion hypothesis we may
deduce that y is in Z[P(C æ)], and thus must be equal to an integral multiple*
* of P(C æ)n-1. But this
is not possible since by Theorem 2.3 the image of eæ*under augmentation is zero*
*, thus so is the image of
y, whereas it is well-known that (P1)n-1 is non-zero in MU* for any n > 0.
Finally, we must estabish the base case, which is for the degree two part of *
*R. Here we want to establish
that if ae-1æ+ be-1æ*is ~(x) for some x, then a = b. By subtracting bP(C æ) f*
*rom x, it suffices to show
that no non-zero integral multiple of e-1æis in the image of ~. But if ~(z) = c*
*e-1æ, then ~(eæz) = c, so that
eæz = c by Theorem 2.2, which implies that 0 = c once we apply the augmenation *
*map to the equality.
Now we proceed with the computation of MUSF*, which follows from Theorem 2.4 *
*by noting that any
class in MUS1*is in MUSF*if and only if the only representations which appear i*
*n its definition are æ
and æ*. For these representations, we have K(æ) and K(æ*) are the trivial subg*
*roup of S1 and thus
the idempotents fiæ and fiæ* project onto the split image of MU* in MUT*. In t*
*he "stable manifolds"
interpretation of Bröcker and Hook, fiæ and fiæ*take a class M ! D(V ) and impo*
*se a trivial S1-action on
both M and D(V ). Let _xdenote fiæ(x).
Definition 3.5. Let B be the set of MUT*elements {eæ, eæ*, Zn,æ, and Zn,æ*} whe*
*re n 2. Order B by
the degree of the classes, with the additional needed convention that Zn,æ< Zn,*
*æ*and eæ < eæ*.
BORDISM OF SEMI-FREE S1-ACTIONS *
* 5
Theorem 3.6. MUSF*is generated as a ring by classes iæ jæ*(x) where x 2 B and *
*if x = eæ*, j = 0.
Relations are
(1)eæ æ(x) = x - ~x= eæ*_æ*(x),_
(2) æ*æ(x) = ææ*(x) + æ(x) ææ*(eæ),
(3) V(x)(y - ~y) = (x - ~x) V(y), where V is æ or æ*,
(4) ææ*(eæ) = æ*æ(eæ*).
Additionally, we require the calculuations ~eV= 0 and V(eV) = 1, where V is æ *
*or æ*.
An additive basis over MU* is given by monomials iæ jæ*(x)m where x 2 B and *
*m is a monomial in
the y x in B.
Proof.The computation of ring structure is an immediate application of Theorem *
*2.4, using the fact that
any class in MUT*is in MUSF*if and only if the only representations which appea*
*r in its definition are æ
and æ*. *
* Q
The identification of the additive basis follows from the fact that one may t*
*ake any product ikæ jkæ*(xk)
where xk 2 B and use relation 3 repeatedly to reduce to a sum of monomials such*
* that only the minimal
element of B appearing in each monomial is operated on by any æ or æ*. Then, *
*one uses relation 2 to
reorder these operations so that the æ*are applied before the æ.
We now turn to a computation of SF*by proving the version of Conjecture 2.7 *
*for semi-free actions.
We start by making geometric constructions of æ and æ*on classes represented *
*by honest G- manifolds.
These constructions follow ones made by Conner and Floyd (hence the name given *
*to the general opeations
V).
Lemma 3.7. ~( æ(x)) = e-1æ(~(x) - ~x) and similarly ~( æ*(x)) = e-1æ*(~(x) - ~x*
*).
Definition 3.8. Define fl(M) for any stably complex S1-manifold to be the stabl*
*y complex S1-manifold
__
fl(M) = M xS1S3 t (-M ) x P(C æ),
where S3 has the standard Hopf S1-action and the S1-action on M xS1S3 is given *
*by
(1) i . [m, z1,=z2][i . m, z1,.iz2]
Define ~fl(M) similarly with the quotient of M x S3 now being by the S1 action *
*in which ø sends m, (z1, z2)
to øm, (øz1, ø-1z2) and with induced S1 action on the quotient given by
-1
(2) i . [m, z1,=z2]i . m, z1, i. z2
Proposition 3.9. Let M be a stably complex S1-manifold. Then æ[M] = [fl(M)] an*
*d æ*[M] = [~fl(M)]
in MUS1*.
Proof.By Lemma 3.7 and the injectivity of ~, it suffices to check the fixed set*
*s of fl(M) and ~fl(M).
One set of fixed points of fl(M) are points [m, z1, z2]such that m is fixed in *
*M and z2 = 0. This fixed
set is diffeomorphic to MG , and its normal bundle is the normal bundle of MG i*
*n M crossed with the
representation æ. In the localization, crossing with æ coincides with multiplyi*
*ng by e-1æ. The second set of
fixed points are [m, z1, z2]such that z1 = 0. This set of fixed points is diffe*
*omorphic to M, and its normal
bundle is the trivial bundle æ-1. __ *
* __
Hence, if x = ~([M]), then_the image of [fl(M)] is xe-1æ+M e-1æ-1. By subtrac*
*ting the image of M xP(C æ)
we are left with xe-1æ- M e-1æ, which by Lemma 3.7 is ~( æ([M])).
The analysis is similar for ~fl(M).
Thus, the classes I(P(Cn æ)) and I(P(Cn æ*)) can be realized geometrica*
*lly. Along similar lines
we have the following.
Lemma 3.10. ææ*(eæ) = P(C æ).
6 DEV SINHA
Proof.The equality of these classes follows from computation of their image und*
*er ~. Following the
methods of tom Dieck [4] as applied in Proposition 4.14 of [11], the isolated f*
*ixed point of P(C æ)with
normal bundle æ contributes a term of e-1æto its image under ~, and similarly t*
*he other fixed point
contributes an e-1æ*. Thus, ~(P(C æ)) = e-1æ+_e-1æ*._To show that this is als*
*o ~( æ æ*(eæ)), by applying
Lemma 3.7 twice it suffices to compute that æ*(eæ)= -1. This equality in turn *
*follows from giving a
Bröcker-Hook model for æ*(eæ) as D(æ*) mapping to D(æ) through complex conjuga*
*tion, which forgetting
S1-action has degree -1.
We are now ready to prove the semi-free case of the geometric realization con*
*jecture.
Theorem 3.11. The following square is a pull-back square
SF* ----! MUSF*
?? ?
y ~?y
PSF*= MU*[e-1æ, e-1æ*, Zn,æ,-Zn,æ*]---!MU*[eæ1, eæ1*, Zn,æ, Zn,æ**
*].
Proof.We first go through the list of generators of MUSF*, determine which map *
*to P*, and show that
those which do have geometric representatives.
By Proposition 3.9, the iæ jæ(x) where x = P(Cn æ) or P(Cn æ*) are in S*
*F*.
Next, by Lemma 3.7,
X ______
(3) ~( iæ(eæ*)) = eæ*e-iæ+ iæ(eæ*)e-m-iæ,
which is not in PSF*as eæ*appears with a positive power.
This leaves iæ jæ*(eæ), which if i = 0 has image under ~ which is not in P* *
*by a computation as in
Equation 3. For i > 0, note that relation 2 of Theorem 3.6 says that æ æ*(x) *
*= æ* æ(x) modulo
MU*[P(C æ)], which is of course in SF*. Thus, modulo SF*, we have
iæ jæ*(eæ) = i-1æ j-1æ*( æ æ*(eæ)),
which by Lemma 3.10 is i-1æ j-1æ*(P(C æ)). Again applying Proposition 3.9, t*
*his class is in SF*.
Theorem 3.6 gave an additive basis for MUSF*as given by monomials M = iæ jæ**
*(x)m where x 2 B =
{eæ, eæ*, P(Cn æ), P(Cn æ*)} and m is a monomial in the y x in B. If x = *
*eæ and i = 0 or if x = eæ*
and j = 0, then by a computation using Lemma 3.7 as above, ~(M) =2PSF*, regardl*
*ess of what m is, as
eæ (respectively eæ*) will appear with a positive power in the leading term of *
*~(M). Otherwise, M is a
product of generators which we have shown are in SF*.
From the proof of Theorem 3.11, an explicit computation of SF*including geom*
*etric representatives is
immediate, since SF*is just the sub-ring of MUSF*generated by the iæ jæ*P(Cn *
*æ) and iæ jæ*P(Cn æ*)
for n 1. Given the general complexities of equivariant bordism, in particular*
* for the geometric theories,
SF*has a remarkably simple form.
Theorem 3.12. SF*is generated as an algebra over MU* by the classes fliflj*P(C*
*n æ) and fliflj*P(Cn æ*)
where n 1. Relations are
(1)fl(x)(y - ~y) = (x - ~x)fl(y),_and similarly for fl*,
(2)fl*fl(x) = flfl*(x) + fl(x)P(C æ).
BORDISM OF SEMI-FREE S1-ACTIONS *
* 7
References
[1]M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology *
*23 (1984) 1-28.
[2]T. Bröcker and E. Hook, Stable equivariant bordism. Math. Z. 129 (1972), 26*
*9-277.
[3]P.E. Conner and E.E. Floyd, Differentiable Periodic Maps. Springer, Berlin-*
*Heidelberg-New York, 1964.
[4]T. tom Dieck, Bordism of G-manifolds and integrality theorems. Topology 9 (*
*1970), 345-358.
[5]J. Duistermaat and G. Heckman, On the variation in the cohomology of the sy*
*mplectic form of the reduced phase space.
Invent. Math. 69, 259-268.
[6]F. Kirwan, The cohomology of quotients in symplectic and algebraic geometry*
*. Princeton University Press, 1984.
[7]J.P. May et. al., Equivariant Homotopy and Cohomology Theory. Volume 91 of *
*the CBMS Regional Conference Series
in Mathematics. AMS Publications, Providence, 1996.
[8]D. McDuff, The moment map for circle actions on symplectic manifolds. J. Ge*
*om. Phys. 5, 149-160 (1988).
[9]D. Metzler Cohomological localization for manifolds with boundary. Int. Mat*
*h. Res. Not. (2002) 24, 1239-1274.
[10]S. Tolman, and J. Weitsman On semifree symplectic circle actions with isola*
*ted fixed points. Topology 39 (2000), no. 2,
299-309.
[11]D. Sinha Computations of complex equivariant bordism rings. Amer. Journal o*
*f Math. 123 (2001), 577-605.
[12]D. Sinha The geometry of the local cohomology filtration in equivariant bor*
*dism. Homology Homotopy Appl. 3 (2001),
no. 2, 385-406