The signature and the elliptic genus of π_{2}-finite manifolds with circle actions

Haydeé Herrera^{∗} and Rafael Herrera^{†}^{‡}

Abstract

We prove the vanishing of various characteristic numbers, such as the signature and the Â-genus, on manifolds with finite second homotopy group and which admit smooth S^{1}-actions. More precisely, we prove the vanishing of various coefficientes of the elliptic genus on non-spin π_{2}-finite manifolds when the S^{1}-action either satisfies a “parity” condition or has isolated fixed points only.

1 Introduction

We shall call a manifold with finite second homotopy group a π_{2}-finite manifold. Our interest in the topology of these manifolds arose from geometrical considerations that can be found in [9, 10, 13, 16]. We have found that they have several features remarkably similar to those of spin manifolds and, in fact, this paper elaborates further in this direction. These features become particularly interesting when observed on non-spin π_{2}-finite manifolds on which the Dirac operator cannot be defined globally. Nevertheless, the Â-genus vanishes on non-spin π_{2}-finite manifolds with circle actions. Note that in this case, the Â-genus no longer represents the index of an elliptic operator and, in principle, is only a rational number [8].

In Section 2, we define odd and even S^{1}-actions on oriented manifolds, and prove that every circle action on a π_{2}-finite manifold is either even or odd. Given this fact, we prove various vanishing results for characteristic numbers such as the following statement (see Theorem 2.1 (i))

Let M be an oriented, connected, compact π_{2}-finite manifold, |π_{2}(M)| < ∞. Assume M admits an odd smooth S^{1}-action. Then the signature of M vanishes

sign(M) = 0.

∗Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA.

E-mail: haydeeh@camden.rutgers.edu

†Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544, USA, and Mathematics Department, The City University of New York, College of Staten Island, 2800 Victory Blvd., Staten Island, NY 10314, USA. E-mail: rherrera@math.princeton.edu

‡Partially supported by a Guggenheim Fellowship, NSF grant DMS-0204002, and PSC-CUNY grant #60031-33-34.

The elliptic genus Φ of a compact, oriented, smooth 2n-dimensional manifold M can be defined as the index of the signature operator d_{s} twisted in the following way

Φ(M) = ind (d_{s} ⊗ ⨂_{i=1}^{∞} ⋀_{q^{i}}T ⊗ ⨂_{i=1}^{∞} S_{q^{i}}T ) ,  (1)

where T = T M_{c} is the complexified tangent bundle, S_{a}T = ∑_{j=0}^{∞} a^{j} S^{j}T, ⋀_{a}T = ∑_{j=0}^{∞} a^{j} ⋀^{j}T, and S^{j}T, ⋀^{j}T denote the j-th symmetric and exterior tensor powers of T respectively (c.f. [17, 18, 11]). The first few terms of the series of bundles are

R_{0} = 1
R_{1} = 2T
R_{2} = 2(T^{⊗2} + T)
R_{3} = 2T^{⊗3} + ⋀^{3}T + S^{3}T + 4T^{⊗2} + 2T, …

so that the constant term in Φ(M) is sign(M). Witten conjectured the rigidity of (1) under S^{1}-actions on spin manifolds [17, 18]. The rigidity theorem was proved by Taubes [15], Bott and Taubes [3].

In [8, 10], we proved the vanishing of the Â-genus and the rigidity of the elliptic genus under S^{1}-actions on non-spin π_{2}-finite manifolds, which show that these nonspin manifolds are similar to spin manifolds with respect to S^{1}-actions. As mentioned above, we elaborate further on this similarity by proving the vanishing of the elliptic genus under various conditions on the action.

In Section 3 we assume that the S^{1}-action has isolated fixed points only and derive further vanishing theorems.

Acknowledgements. The authors wish to thank the referee for useful remarks. The first named author wishes to thank Loring Tu for his encouragement and valuable discussions, and the Institute for Pure and Applied Mathematics (UCLA) for its hospitality during the Symplectic Geometry and Physics Program. The second named author wishes to thank the Max-Planck-Institut für Mathematik for its hospitality and support.

2 Even and odd S^{1}-actions

Let M be an oriented, connected, compact 2n-dimensional manifold. Assume M is endowed with a (non-trivial) smooth S^{1}-action. Let M^{S^{1}} denote the fixed point set of the circle action. At each point p ∈ M^{S^{1}}, the tangent space of M splits as a sum of S^{1} representations

T_{p}M = T_{p}M^{S^{1}} ⊕ L^{m_{1}} ⊕ … ⊕ L^{m_{k}},

where L^{a} denotes the S^{1} representation on which λ ∈ S^{1} acts by multiplication by λ^{a}. The space T_{p}M^{S^{1}} is a trivial representation of S^{1}.

The numbers m_{1}, …, m_{k} are called the exponents (or weights) of the S^{1}-action at the (connected component of M^{S^{1}} containing the) point p. They are not canonical and their sign can be changed in pairs. Consider the sum of the exponents S(p) = ∑_{i=1}^{k} m_{i} at p ∈ M^{S^{1}}. The number S(p) is constant on the same connected component of M^{S^{1}}, but may vary for different connected components.

Example. Consider the S^{1} action on ℂℙ^{2} given by

λ · [x_{0} : x_{1} : x_{2}] ⟶ [x_{0} : λx_{1} : λ^{2}x_{2}],

where λ ∈ S^{1}. The (isolated) fixed points for this action are l_{0} = [1 : 0 : 0], l_{1} = [0 : 1 : 0] and l_{2} = [0 : 0 : 1]. The exponents of the S^{1}-action at each point are:

\begin{matrix}l_{0} & 1 & 2 & S(l_{0}) = 3 \\ l_{1} & −1 & 1 & S(l_{1}) = 0 \\ l_{2} & −2 & −1 & S(l_{2}) = −3 \end{matrix}

Notice that the numbers S(l_{i}) do not even have the same parity.

On the other hand, there are classes of manifolds with S^{1}-actions whose numbers S(p) have the same parity for every fixed point p ∈ M^{S^{1}}. For example, spin manifolds satisfy such a condition [1], as well as π_{2}-finite manifolds (c.f. [8, 10], see below). We may assume that M^{S^{1}} is non-empty since otherwise M is rationally zero-bordant and Φ(M) = 0.

Definition 2.1 A circle action on an oriented 2n-dimensional manifold M with non-empty fixed point set will be called even if the sum S(p) ≡ 0 (mod 2) for all p ∈ M^{S^{1}}, and odd if S(p) ≡ 1 (mod 2) for all p ∈ M^{S^{1}}.

In the next few paragraphs we verify that S^{1}-actions on π_{2}-finite manifolds are either odd or even, and that the codimension of the fixed point submanifolds behave as in the spin case [1].

Lemma 2.1 Let M be an oriented, connected, compact 2n-dimensional manifold with finite second homotopy group, |π_{2}(M)| < ∞. Assume M admits a smooth S^{1} action. Then S(p_{1}) = S(p_{2}) for all p_{1}, p_{2} ∈ M^{S^{1}}. In particular, the S^{1}-action is either even or odd.

Proof. Consider the S^{1}-decompositions of the tangent space at the two points in terms of generator L of the representation ring R(S^{1}) = ℤ{L} (the ring of finite Laurent series in L)

T_{p_{1}}M = L^{m_{1}} ⊕ … ⊕ L^{m_{n}},
T_{p_{2}}M = L^{m′_{1}} ⊕ … ⊕ L^{m′_{n}},

where the m_{i} and m′_{i} are the exponents of the S^{1} action at each point. By [4], the virtual representation T_{p_{1}}M − T_{p_{2}}M can be factored as follows

T_{p_{1}}M − T_{p_{2}}M = (1 − L)^{2} ⊗ (⨁_{j} b_{j}L^{j}) ,  (2)

where the set {b_{j} ∈ ℤ} is finite. Clearly

S(p_{1}) − S(p_{2}) = ∑_{j} b_{j} · j − 2 ∑_{j} b_{j} · (j + 1) + ∑_{j} b_{j} · (j + 2) = 0.

□

Lemma 2.1 allows us to talk about even and odd actions on π_{2}-finite manifolds in a similar fashion to that of the spin case.

Proposition 2.1 Let M be an oriented, connected, compact 2n-dimensional manifold with finite second homotopy group, admitting a smooth S^{1}-action. Let ℤ_{2} = {±1} be the subgroup of S^{1} generated by the involution.

Let X be a connected component of the ℤ_{2}-fixed point set M^{ℤ_{2}} such that X ∩ M^{S^{1}} ≠ ∅. Then

codim(X) ≡ 0 (mod 4) if the action is even,
codim(X) ≡ 2 (mod 4) if the action is odd.

Proof. Let p ∈ X ∩ M^{S^{1}} and consider the S^{1} decomposition of

T_{p}(M) = L^{m_{1}} ⊕ … ⊕ L^{m_{n}},

where m_{i} ∈ ℤ. This (complex) decomposition reduces to twice the (real) decomposition under the subgroup ℤ_{2}. The summands with even m_{i} reduce to twice the trivial real representation of ℤ_{2}, and the ones with m_{i} odd reduce to twice the non-trivial representation. Therefore, half the codimension of X is

\frac{1}{2} codim(X) = ∑_{m_{i} odd} 1 ≡ ∑_{m_{i} odd} m_{i} (mod 2)

≡ ∑_{i=1}^{2n} m_{i}  (mod 2)

≡ \{ \begin{matrix}0 (mod 2) & if the action is even \\ 1 (mod 2) & if the action is odd \end{matrix}
□

From now on, we shall assume that dim(M) ≡ 0 (mod 4).

Theorem 2.1 Let M be an oriented, connected, compact 4n-dimensional manifold with finite second homotopy group, |π_{2}(M)| < ∞. Assume M admits a (non-trivial) smooth S^{1}-action, and let ℤ_{2} = {±1} the subgroup generated by the involution −1 ∈ S^{1}.

(i) If the action is odd, then the elliptic genus and the signature of M vanish

Φ(M) = 0  and  sign(M) = 0.

(ii) If the action is even and codim(Y) ≥ 4r for all the connected components Y of M^{ℤ_{2}} that contain S^{1}-fixed points, then the characteristic numbers Â(M, R′_{j}) vanish for 1 ≤ j ≤ r −1. If r ≥ n/2 then Φ(M) does not depend on the variable q and Φ(M) = sign(M). If r > n/2, then the elliptic genus and the signature of M vanish

Φ(M) = 0  and  sign(M) = 0.

(iii) In particular [8], the Â-genus of M vanishes regardless of the parity of the action

Â(M) = 0.

Remark. Note that the vanishing of characteristic numbers such as Â(M) and Â(M, T) is new and highly non-trivial on non-spin π_{2}-finite manifolds, since they do not represent the index of an elliptic operator.

Proof. The main tool for the proof is the rigidity theorem of the elliptic genus for π_{2}-finite manifolds with S^{1}-actions [8, 10]. Namely, let G be a Lie group acting on an oriented, smooth manifold by diffeomorphisms that preserve the orientation. The coefficients of q^{k} in Φ(M) are of the form sign(M, R_{k}) where the bundle R_{k} is such that the G-action lifts to it: R_{0} = 1, R_{1} = 2T, R_{2} = 2(T ⊕ T ⊗ T), etc. Thus, for g ∈ G, the equivariant signature sign(M, R_{k})_{g} is defined for all k, and we get an equivariant elliptic genus

Φ(M)_{g} = ∑_{k=0}^{∞} sign(M, R_{k})_{g} · q^{k},

which is also a power series in q. The rigidity theorem of the elliptic genus for π_{2}-finite manifolds states that

Φ(M) = Φ(M)_{g}  (3)

for all g ∈ G. Notice that this is certainly not the case for manifolds with infinite second homotopy group, such as the complex projective plane ℂℙ^{2}.

Recall that Φ(M) has the following expansion at the other cusp [11]

Φ̃(M) = \frac{1}{q^{n/2}} ∑_{j=0}^{∞} Â(M, R′_{j}),

where R′_{j} is a sequence of representations in terms of T starting with R′_{0} = 1, R′_{1} = −T, R′_{2} = ⋀^{2}T + T, etc., and Â(M, R′_{j}) denotes the Â-genus twisted by the bundle R′_{j}. This (rational) characteristic number is defined for all oriented compact manifolds. It constitutes the index of the Dirac operator with coefficients in R′_{j} only when the manifold is spin, in which case it is an integer.

By the detailed analysis of [7], which is also valid in our case due to the rigidity theorem [8, 10], we can consider only those components of M^{g} of M^{ℤ_{2}} that contain S^{1}-fixed points. Let M^{g} ∘ M^{g} denote its (transversal) self-intersection. M^{g} ∘ M^{g} is oriented and smooth. By [12]

Φ(M)_{g} = Φ(M^{g} ∘ M^{g}).

Hence, by the rigidity (3)

Φ(M) = Φ(M^{g} ∘ M^{g}),

which implies

Φ̃(M) = Φ̃(M^{g} ∘ M^{g}).  (4)

The codimension of M^{g} is positive and even, therefore the right hand side of (4) has a pole of order less than n/2, which implies that the coefficient of q^{n/2} in the left hand side of (4) must vanish, proving (iii).

If the action is odd, codim(M^{g} ∘ M^{g}) ≡ 4 (mod 8). The order of the pole for Φ̃(M) in q is both integral and half integral in (4). Hence, Φ(M) = 0, proving (i).

If the action is even the hypotheses on the codimension indicates only that the order of the pole in Φ̃(M^{g} ∘ M^{g}) is smaller than the one in Φ̃(M) according to the codimension Hence the vanishing of the coefficients in (ii) □

3 Isolated fixed points

Theorems 2.1 shows that both the type of the action and the codimension of fixed point sets of the involution −1 ∈ S^{1} can determine the elliptic genus and the signature of a π_{2}-finite manifold. In this section, inspired by [6, Section 6.1], we shall study the case when the S^{1} action has isolated fixed points only. Again, the main tool will be the rigidity theorem of the elliptic genus on π_{2}-finite manifolds.

Let M be an oriented, compact, connected, π_{2}-finite 2n-dimensional manifold admitting a smooth S^{1}-action with isolated fixed points only. Let F = M^{S^{1}} be the fixed point set and {m_{p,i} : i = 1...n} the set of exponents of the action at each p ∈ F. Notice that since p is isolated, all the exponents are non-zero.

By [2], the equivariant elliptic genus is

Φ(M)_{S^{1}}(λ) = ∑_{p∈F} μ_{p}(q, λ)

where λ is a topological generator of S^{1}. The local contributions of the fixed points are

μ_{p}(q, λ) = ∏_{i=1}^{n} \frac{1 + λ^{−m_{p,i}}}{1 − λ^{−m_{p,i}}} · ∏_{l=1}^{∞} \frac{(1 + q^{l}λ^{m_{p,i}})(1 + q^{l}λ^{−m_{p,i}})}{(1 − q^{l}λ^{m_{p,i}})(1 − q^{l}λ^{−m_{p,i}})}.  (5)

Since the elliptic genus is rigid on π_{2}-finite manifolds with S^{1}-actions, Φ(M)_{S^{1}}(λ) does not depend on λ.

Let

ψ(τ, z) = \frac{1 + λ^{−1}}{1 − λ^{−1}} · ∏_{l=1}^{∞} \frac{(1 + q^{l}λ)(1 + q^{l}λ^{−1})}{(1 − q^{l}λ)(1 − q^{l}λ^{−1})}

where q = e^{2πiτ} and λ = e^{2πiz}. The function ψ(τ, z) is meromorphic on H × ℂ with zeroes of order one at ℤ · τ + ℤ − 1/2 and poles of order one at ℤ · τ + ℤ. Define

φ(τ, z) = ∑_{p∈F} ∏_{i=1}^{n} ψ(τ, m_{p,i} · z).  (6)

Thus,

Φ(M) = φ(τ, z),

by the aforementioned rigidity, and it does not depend on z.

In order to derive more detailed information on the signature, we shall borrow some observations from A. Dessai [6, Section 6.1]. The function ψ is related to the Weierstrass Φ-function [11] and has the following properties:

•  ψ(τ, −z) = −ψ(τ, z),  ψ(τ, z + 1) = ψ(τ, z),
ψ(τ, z + 1/2) = ψ(τ, z)^{−1}, ψ(τ, z + τ) = −ψ(τ, z),
ψ(τ, 1/2) = 0,  ψ(τ, 1/4) = \frac{1 − i}{1 + i},  ψ(τ, 3/4) = \frac{1 + i}{1 − i},

• If a ∈ ℤ is odd, ψ(τ, a/4) is constant in τ.

• If a ∈ ℤ is such that a ≢ 0 (mod 3),

ψ(τ, (aτ)/3) = ±ψ(τ, τ/3)

• If s, t ∈ ℤ, such that s ≡ 1 (mod 2) and t ≡ 0 (mod 2) then

(ψ(τ, s · z) · ψ(τ, t · z))|_{z=1/2} = \frac{s}{t}

is a constant function in τ.

• If s, t ∈ ℤ, such that s ≡ 2 (mod 4) and t ≡ 0 (mod 4) then

(ψ(τ, s · z) · ψ(τ, t · z))|_{z=1/4} = \frac{s}{t}

is a constant function in τ.

For each p ∈ F let

m(p)_{e} = #{m_{p,i} ≡ 0 (mod 2)}, m(p)_{o} = #{m_{p,i} ≡ 1 (mod 2)},
m̃(p)_{e} = #{m_{p,i} ≡ 0 (mod 4)},  m̃(p)_{o} = #{m_{p,i} ≡ 2 (mod 4)}.

The following theorem follows from the properties of ψ.

Theorem 3.1 Let M be an oriented, compact, connected, π_{2}-finite 2n-dimensional manifold admitting a smooth S^{1}-action with isolated fixed points only. Then 1. If for every p ∈ F

m(p)_{e} ≤ m(p)_{o},  then  Φ(M) = sign(M).

If the inequality is strict, Φ(M) = sign(M) = 0.

2. If for every p ∈ F

m̃(p)_{e} ≤ m̃(p)_{o},  then  Φ(M) = sign(M).

If the inequality is strict, Φ(M) = sign(M) = 0.

3. If m_{p,i} ≢ 0 (mod 3) for all p ∈ F and all i, then Φ(M) = 0.

4. If dim(M) ≡ 4 (mod 8) and Φ(M) = sign(M), then sign(M) = 0.

Part 4 follows from modular properties of the elliptic genus. □

Corollary 3.1 Let M be as in Theorem 3.1. Assume that the S^{1}-action is also semi-free. Then all the exponents of the action are ±1 so that

Φ(M) = sign(M) = 0.
□

Corollary 3.2 Let M be as in Theorem 3.1. Assume that the S^{1}-action can be extended to a SU(2) action with M^{SU(2)} = M^{S^{1}}. Then, all the exponents of the action are odd and

Φ(M) = sign(M) = 0.
□

Finally, let us consider manifolds with finite even homotopy groups.

Theorem 3.2 Let M be an oriented, compact, connected, 2n-dimensional manifold with |π_{2i}(M)| < ∞ for all 1 ≤ i ≤ dim(M)/2. Then M admits no smooth S^{1}-action with isolated fixed points only. In particular, if M is the underlying manifold of a compact connected Lie group, then M does not admit smooth S^{1}-actions with isolated fixed points only.

Proof. By [5], if |π_{2i}(M)| < ∞ for 1 ≤ i ≤ dim(M)/2, the fixed point set M^{S^{1}} is connected, which in the isolated fixed point case means there is only one fixed point. A smooth S^{1}-action on an oriented manifold, however, cannot have exactly one fixed point. □

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