Mod $p$ vanishing theorem of Seiberg-Witten invariants for 4-manifolds with $Z_p$-actions

N. Nakamura

REVIEW

Mod $p$ vanishing theorem of Seiberg-Witten invariants for 4-manifolds with $Z_p$-actions

N. Nakamura

Differential Geometry · Submitted 28 Feb 2006 · 22 references

Pith's verdict: Solid extension of Fang's mod p vanishing to b1>0; the main gap is an asserted, unproved equality between the finite-dimensional zero count and the genuine Seiberg-Witten invariant.

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CitationN. Nakamura. "Mod $p$ vanishing theorem of Seiberg-Witten invariants for 4-manifolds with $Z_p$-actions." arXiv:math/0602654 (2006). https://pith.science/paper/XFB2ZSSF

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Editor's note

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Short version: this is a real extension of Fang's theorem, not a repackaging. The geometric proof via dimension counting of fixed sets is clean, and the examples against known SW values (Witten, Turaev torsion) give independent support. The theorem likely holds. But there is one gap a referee must push on.

What's new: Theorem 1.2 extends Fang's b1=0 vanishing theorem to b1≥1; Theorem 1.8 handles odd-type Z2 actions; Section 4 gives analogous results for cut-down invariants using tori in the Jacobian. The proof method, Lemma 3.1's fixed-point-set computation, is a genuine alternative to Fang's equivariant K-theory. The inequality (1.3) is exactly the dimension condition, and the argument in §3(ii)-(iii) is sound assuming the reduction to the finite-dimensional picture.

The soft spot: the paper never proves that the signed count of zeros of the finite-dimensional approximation f^{-1}(0)/U(1) equals the genuine Seiberg-Witten invariant. §2(iii) calls it an abuse and asserts the fundamental classes agree after perturbation, but that is precisely what needs an argument or a citation. The finite-dimensional map depends on the cutoff λ and is a map between different bundles; it is not a routine corollary of Furuta's lemma. If that bridge fails, the divisibility argument shows nothing about SW_X(c). The stress-test note is right to flag this. It is a single gap, but it is the central one. A referee should ask for a proof, or an exact reference to where this comparison is established.

Minor issues: G-equivariant transversality is described in a few lines; it is probably standard but needs detail. The ad hoc definition of coefficients when X^G=∅ is only used to show no partition satisfies (1.3), so it does not weaken the theorem. The paper honestly says there is no application for b1>0 yet; that limits significance but not correctness.

Bottom line: the mathematical core is credible, and the examples checking against known invariants make me think the result is true. With the finite-dimensional comparison proved or cited, this is a solid paper for a specialist journal. As is, it deserves peer review and a referee who knows the Bauer-Furuta/Furuta setup well enough to close that gap.

Referee report

Significance

The core dimension count is clean and checkable: Lemma 3.1 correctly identifies the fixed-point set of the U(1)-quotient, and equations (3.2)-(3.4) convert (1.3) exactly into the requirement that the fixed strata have dimension smaller than the bundle rank over them. The coefficients k_j^l are computed from the Atiyah-Segal G-index formula, not adjusted to force the conclusion, and Lemma 3.5 correctly shows that the X^G = empty case is vacuous for b_1 >= 1. The examples are a genuine strength: for X = T^2 x Sigma_{3h}, Theorem 1.2 (with all k_j^l = 0) predicts SW cong 0 mod 3, and the independent computation SW = +/- C(6h-2, 3h-1), obtained both from Witten's factorization argument and from Turaev torsion, confirms it; Remark 5.22 even exhibits a nearby case where (1.3) fails and vanishing does not occur. If the gap in the finite-dimensional comparison identified in Major Comment 1 is closed, Theorem 1.2 is a worthwhile extension of Fang's theorem. The paper is candid about its limitations: it states that no application for b_1 > 0 has been found yet, and it relies on the unpublished preprint [13] for the approximation construction.

Major objections (2)

[Section 2(iii), paragraph beginning 'First, note that the moduli space...'] The stress-test concern lands exactly here. The equality asserted in Section 2(iii) - 'after perturbation, the fundamental class of f_0^{-1}(0)/U(1) is equal to that of the perturbed moduli space' - is the only bridge between the zero set of the finite-dimensional approximation, whose free G-action is used in Sections 3(ii)-(iii), and the genuine invariant SW_X(c) = <U^{d(c)/2}, [M_c]>. Lemma 2.2 does not imply this equality: it controls the linear part of the map and the compactness of the zero set only. What is needed is a parameterized (over J), G-equivariant version of the Bauer-Furuta comparison theorem, and it must preserve the evaluation of U^d and of the classes pulled back from the Jacobian under the induced cobordism; no proof or precise reference for the equivariant family version is given. Since every divisibility conclusion in Section 3 is drawn on the finite-dimensional zero set, this missing comparison is load-bearing for Theorem 1.2 and must be supplied, or the exact statement in the literature must be cited and adapted.
[Sections 3(ii)-(iii), after equations (3.2)-(3.4)] The G-equivariant perturbation step is asserted rather than proved. The section f-bar obtained from Section 2(ii) is only continuous - Lemma 2.2 supplies continuous homomorphisms built from smoothed spectral projections - so the claims 'we can perturb the section f-bar ... so that f-bar has no zero on V-bar^G' and 'it is easy to carry out a G-equivariant perturbation ... so that f-bar is transverse to the zero section' require an equivariant transversality argument (e.g., Wasserman's theorem) or a smoothing argument that leaves the zero count unchanged. The parenthetical about quotient spaces covers only the free part of the action, not the fixed strata where the dimension inequality (1.3) is the relevant input. Because the mod-p count presupposes a 0-dimensional zero set that is a manifold, this is also a load-bearing point and should be proved or referenced.

Minor objections (6)

[Theorem 4.2] The inequality in Theorem 4.2 is stated for j = 1,...,p-1 only, while the analogous Theorem 4.1 and the fixed-stratum computation of Sections 3(i)-(iii) use the condition for all j, including j = 0; the stratification argument in the proof of Theorem 4.2 does not explain why the j = 0 component needs no dimension condition, so the statement should be justified or amended.
[Proof of Theorem 4.2] The proof of Theorem 4.2 contains corrupted notation: 'outsideVT2' should refer to the locus V-bar restricted to T_2, and the expression 'V-bar|T2\T2' does not parse; please rewrite this paragraph with consistent notation for the strata.
[Section 5(ii), Proposition 5.8] In the even-type (free-action) case of Proposition 5.8 the proof says only that the claim is 'obvious from the Lefschetz formula and the G-signature formula'; one sentence of detail would make the argument checkable.
[Section 3(ii)] The sentence 'the number of elements in the moduli space is a multiple of p' should read 'the signed count of points of the moduli space is a multiple of p'; the conclusion is unaffected since a free action multiplies the signed count by p (or by zero in the orientation-reversing Z_2 case), but the invariant is a signed count.
[Sections 2(ii)-(iii) and 3] The notation for the finite-dimensional map is inconsistent: Section 2(ii) writes Psi_f = D_f + Q_f, Section 2(iii) writes f_0 = Psi_f, and Section 3 uses f = f_0 + v for the perturbed map; please fix the notation at first use.
[Section 3(iv)] The sentence 'the assumption X^G is nonempty can be omitted logically' should refer explicitly to Lemma 3.5: for b_1 >= 1 and X^G = empty, no partition satisfying (1.3) exists, so the hypothesis of Theorem 1.2 is vacuous in that case.

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The paper

The paper claims

A cyclic group action of prime order forces Seiberg-Witten invariants to vanish modulo p when a Dirac-index inequality holds.

The reading

The paper proves a modulo-$p$ vanishing theorem for Seiberg-Witten invariants of smooth 4-manifolds carrying a cyclic group action of prime order. The main result says that if a numerical inequality holds, involving the weights of the equivariant Dirac index over the fixed Jacobian components and the fixed Betti numbers, then the Seiberg-Witten invariant is divisible by $p$. This generalizes an earlier $b_1=0$ vanishing theorem to arbitrary $b_1$, using a geometric argument based on the finite-dimensional approximation of the monopole map rather than equivariant $K$-theory. The relevance is that it gives a computable criterion for the moduli space to admit a free group action, with consequences for the classification of smooth group actions on 4-manifolds. The paper also proves a mod 2 version for involutions of odd type and verifies the theorem on explicit examples.

What carries the argument

The load-bearing object is the $G$-equivariant finite-dimensional approximation $f:V\to W$ of the monopole map, built from a $G$-equivariant finite-rank bundle approximation over the Jacobian (Lemma 2.2). The mechanism that forces vanishing is a dimension comparison on the fixed-point sets. Lemma 3.1 identifies the fixed-point set of the quotient bundle $\bar V=((V_C\setminus\{0\})\times_J V_R)/U(1)$ over a fixed component $J_l$ as a disjoint union over weights $j$ of $\mathbb{P}(k_j^{+}C_j)\times\mathbb{R}_+$, from which the dimension of each fixed component is $2k_j^{+}-1+a+b_1^G$. The corresponding rank of the cut-down bundle in $\bar E^G$ is $2(k_j^{-}+d_j)+a+b_+^G$. Inequality (1.3) is exactly $\dim\bar V^G_{l,j}<\operatorname{rank}\bar E^G_{l,j}$, which allows a $G$-equivariant perturbation to remove all zeros on the fixed-point set; on the complement $G$ acts freely, so the zero count—and hence the Seiberg-Witten invariant—is divisible by $p$.

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2. Let $G=\mathbb{Z}_p$ for a prime $p$, let $X$ be a smooth closed oriented 4-dimensional $G$-manifold with $b^+\ge 2$ and $b_+^G\ge 1$, and let $c$ be a $G$-equivariant $\mathrm{Spin}^c$-structure whose virtual dimension $d(c)=\frac14(c_1(L)^2-\operatorname{Sign}(X))-(1-b_1+b_+)$ is non-negative and even. Write the restriction of the family $G$-index of Dirac operators to each component $J_l$ of $J^G$ as $\sum_{j=0}^{p-1} k_j^l C_j$ in the representation ring $\mathbb{Z}[t]/(t^p-1)$. If there is a partition $d_0+\cdots+d_{p-1}=d(c)/2$ with $d_j\ge 0$ and $2k_j^l<2d_j+1-b_1^G+b_+^G$ for all $j$ and $l$, then $SW_X(c)\equiv 0\pmod p$. In the $d(c)=0$ case this means that after a $G$-equivariant perturbation the moduli space has no $G$-fixed points, so the group acts freely and the signed count vanishes modulo $p$; in higher dimension the same conclusion follows after cutting down by the bundles $L_j$ associated to the weight-$j$ representations.

Load-bearing premise

The proof depends on identifying the signed zero count of the $G$-equivariant finite-dimensional approximation, after perturbation, with the genuine Seiberg-Witten invariant; the paper does not prove this identification and instead adopts it as a convention when it says "we abuse the term moduli space and the notation $M_c$ for $f^{-1}(0)/U(1)$".

Editorial extensions

If this is right

The earlier vanishing theorem for $b_1=0$ follows directly: under its assumptions $b_1=b_1^G=0$ and $b_+=b_+^G$, so the condition $2k_j\le b_+-1$ implies (1.3) for every partition.
For $b_1>0$ the theorem gives a general modulo-$p$ vanishing criterion: it applies whenever the fixed Betti numbers and the weight-$j$ index coefficients satisfy the inequality for some partition of $d(c)/2$.
For a $\mathbb{Z}_2$-action of odd type with respect to $c$, the analogous inequality on the $\mathbb{Z}_4$-index coefficients forces $SW_X(c)\equiv 0\pmod 2$ (Theorem 1.8).
Subtorus cut-down invariants vanish modulo $p$ under the analogous inequality when the torus is $G$-invariant; for a non-invariant torus the sum of the invariants over the $G$-orbit of the torus vanishes modulo $p$ (Theorems 4.1 and 4.2).
On $T^2\times\Sigma_{3h}$ with the diagonal $\mathbb{Z}_3$-action, all index coefficients are zero and $d(c_0)=0$, so Theorem 1.2 yields $SW(c_0)\equiv 0\pmod 3$, in agreement with the exact value $\pm\binom{2g-2}{g-1}$ for $g=3h$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the proof only uses that the group acts freely on the reduced zero set after removing fixed points, the same dimensional argument should extend to arbitrary finite group actions whose fixed-point strata are labelled by irreducible representations; the prime cyclic case is the simplest setting.
Read contrapositively, the theorem says that any $G$-equivariant structure with $SW_X(c)\not\equiv 0\pmod p$ must violate (1.3) for every partition, giving an explicit numerical obstruction to lifting a given action to a $G$-equivariant spin structure; this could be used to detect locally linear actions that are not smooth on manifolds with $b_1>0$, extending the application noted in the introduct
If the finite-dimensional count is compatible with the genuine Seiberg-Witten invariant, the mod $p$ vanishing should persist for refined invariants such as equivariant stable homotopy or cohomotopy refinements; this is a testable strengthening not proved in the paper.
For knot-surgered manifolds mentioned in Remark 5.21, the theorem predicts that the change in the Seiberg-Witten invariant under the $\mathbb{Z}_3$-action is divisible by 3; carrying out that computation would provide an independent check beyond the examples in Section 5.

Original abstract

We give an alternative proof of the mod $p$ vanishing theorem by F.Fang of Seiberg-Witten invariants under a cyclic group action of prime order, and generalize it to the case when $b_1>0$. Although we also use the finite dimensional approximation of the monopole map as well as Fang, our method is rather geometric. Furthermore, non-trivial examples of mod $p$ vanishing are given.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We give an alternative proof of the mod $p$ vanishing theorem by F.Fang of Seiberg-Witten invariants under a cyclic group action of prime order, and generalize it to the case when $b_1>0$. Although we also use the finite dimensional approximation of the monopole map as well as Fang, our method is rather geometric. Furthermore, non-trivial examples of mod $p$ vanishing are given.

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s0	We give an alternative proof of the mod $p$ vanishing theorem by F.Fang of Seiberg-Witten invariants under a cyclic group action of prime order, and generalize it to the case when $b_1>0$.	contribution 0.97	0.93	0.86	asserted 1.00
s1	Although we also use the finite dimensional approximation of the monopole map as well as Fang, our method is rather geometric.	method 0.93	0.84	0.04	asserted 1.00
s2	Furthermore, non-trivial examples of mod $p$ vanishing are given.	result 0.64	0.92	0.13	asserted 1.00

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References

Works cited (22)

[13]

M. Furuta, Stable homotopy version of Seiberg-Witten invariant , preprint, the Max-Planck-Institut f¨ ur Mathematik (http://www.mpim-bonn.mpg.de/), MPIM1997-110

View on Pith
[1]

M. F. Atiyah and R. Bott, A Lefschetz ﬁxed point formula for elliptic complexes II App lications, Ann. Math. 88 (1968), 451–491

View on Pith 1968
[2]

M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions , Essays on topology and related topics, Memoires d´ edi´ e ` a George de Rham (ed. A. Haeﬂiger and R . Narashimhan), Springer-Verlag (1970), 18–28

View on Pith 1970
[3]

M. F. Atiyah and G. B. Segal, The index of elliptic operators: II , Ann. Math. 87 (1968), 531–545

View on Pith 1968
[4]

M. F. Atiyah and I. M. Singer, The index of elliptic operators: III , Ann. Math. 87 (1968), 546–604

View on Pith 1968
[5]

S. Baldridge, Seiberg-Witten vanishing theorem for S1-manifolds with ﬁxed points , Paciﬁc J. Math. 217 (2004), 1–10

View on Pith 2004
[6]

S. Bauer and M. Furuta, A stable cohomotopy reﬁnement of Seiberg-Witten invariant s: I , Invent. Math. 155 (2004), 1–19

View on Pith 2004
[7]

J. Bryan, Seiberg-Witten theory and Z/2p actions on spin 4-manifolds, Math. Res. Lett. 5 (1998), 165–183

View on Pith 1998
[8]

Y. S. Cho and Y. H. Hong, Seiberg-Witten invariants and (anti-)symplectic involutions , Glasgow Math. J. 45 (2003), 401–413

View on Pith 2003
[9]

F. Fang, Smooth group actions on 4-manifolds and the Seiberg-Witten theory , Diﬀ. Geom. Appl. 14 (2001), 1–14

View on Pith 2001
[10]

F. Fang, Smooth group actions on 4-manifolds and Seiberg-Witten invariants , International J. Math. 9, No.8 (1998) 957–973

View on Pith 1998
[11]

R. Fintushel and R. Stern, Knots, links, and 4-manifolds, Invent. Math. 134 (1998), 363–400

View on Pith 1998
[12]

M. Furuta, Monopole equation and the 11 8 -conjecture, Math. Res. Lett. 8 (2001), 279–291

View on Pith 2001
[14]

X. Liu and N. Nakamura, Pseudofree Z/3-actions on K3 surfaces, preprint. 20 NOBUHIRO NAKAMURA

View on Pith
[15]

N. Nakamura, A free Zp-action and the Seiberg-Witten invariants , J. Korean Math. Soc. 39 (2002), No. 1, 103–117

View on Pith 2002
[16]

Y. Ruan and S. Wang, Seiberg-Witten invariants and double covers of 4-manifolds, Comm. Anal. Geom. 8 (2000), No.3, 477-515

View on Pith 2000
[17]

D. Ruberman and S. Strle, Mod 2 Seiberg-Witten invariants of homology tori , Math. Res. Lett. 7 (2000), 789–799

View on Pith 2000
[18]

M. Szymik, Bauer-Furuta invariants and Galois symmetries , preprint

View on Pith
[19]

C. Taubes, The Seiberg-Witten invariants and symplectic forms , Math. Res. Lett. 1 (1994), 769–796

View on Pith 1994
[20]

V. Turaev, Torsions of 3-dimensional manifolds, Progress in Ma thematics, vol. 208, Birkh¨ auser

View on Pith
[21]

S. Wang, A vanishing theorem for Seiberg-Witten invariants , Math. Res. Lett. 2 (1995), 305–310

View on Pith 1995
[22]

E. Witten, Monopoles and four-manifolds , Math. Res. Lett. 1 (1994), 769-796. Research Institute for Mathematical Sciences, Kyoto unive rsity, Kyoto, 606-8502, Japan E-mail address : nakamura@kurims.kyoto-u.ac.jp

View on Pith 1994

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [13]

M. Furuta, Stable homotopy version of Seiberg-Witten invariant , preprint, the Max-Planck-Institut f¨ ur Mathematik (http://www.mpim-bonn.mpg.de/), MPIM1997-110

View on Pith

[2] [1]

M. F. Atiyah and R. Bott, A Lefschetz ﬁxed point formula for elliptic complexes II App lications, Ann. Math. 88 (1968), 451–491

View on Pith 1968

[3] [2]

M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions , Essays on topology and related topics, Memoires d´ edi´ e ` a George de Rham (ed. A. Haeﬂiger and R . Narashimhan), Springer-Verlag (1970), 18–28

View on Pith 1970

[4] [3]

M. F. Atiyah and G. B. Segal, The index of elliptic operators: II , Ann. Math. 87 (1968), 531–545

View on Pith 1968

[5] [4]

M. F. Atiyah and I. M. Singer, The index of elliptic operators: III , Ann. Math. 87 (1968), 546–604

View on Pith 1968

[6] [5]

S. Baldridge, Seiberg-Witten vanishing theorem for S1-manifolds with ﬁxed points , Paciﬁc J. Math. 217 (2004), 1–10

View on Pith 2004

[7] [6]

S. Bauer and M. Furuta, A stable cohomotopy reﬁnement of Seiberg-Witten invariant s: I , Invent. Math. 155 (2004), 1–19

View on Pith 2004

[8] [7]

J. Bryan, Seiberg-Witten theory and Z/2p actions on spin 4-manifolds, Math. Res. Lett. 5 (1998), 165–183

View on Pith 1998

[9] [8]

Y. S. Cho and Y. H. Hong, Seiberg-Witten invariants and (anti-)symplectic involutions , Glasgow Math. J. 45 (2003), 401–413

View on Pith 2003

[10] [9]

F. Fang, Smooth group actions on 4-manifolds and the Seiberg-Witten theory , Diﬀ. Geom. Appl. 14 (2001), 1–14

View on Pith 2001

[11] [10]

F. Fang, Smooth group actions on 4-manifolds and Seiberg-Witten invariants , International J. Math. 9, No.8 (1998) 957–973

View on Pith 1998

[12] [11]

R. Fintushel and R. Stern, Knots, links, and 4-manifolds, Invent. Math. 134 (1998), 363–400

View on Pith 1998

[13] [12]

M. Furuta, Monopole equation and the 11 8 -conjecture, Math. Res. Lett. 8 (2001), 279–291

View on Pith 2001

[14] [14]

X. Liu and N. Nakamura, Pseudofree Z/3-actions on K3 surfaces, preprint. 20 NOBUHIRO NAKAMURA

View on Pith

[15] [15]

N. Nakamura, A free Zp-action and the Seiberg-Witten invariants , J. Korean Math. Soc. 39 (2002), No. 1, 103–117

View on Pith 2002

[16] [16]

Y. Ruan and S. Wang, Seiberg-Witten invariants and double covers of 4-manifolds, Comm. Anal. Geom. 8 (2000), No.3, 477-515

View on Pith 2000

[17] [17]

D. Ruberman and S. Strle, Mod 2 Seiberg-Witten invariants of homology tori , Math. Res. Lett. 7 (2000), 789–799

View on Pith 2000

[18] [18]

M. Szymik, Bauer-Furuta invariants and Galois symmetries , preprint

View on Pith

[19] [19]

C. Taubes, The Seiberg-Witten invariants and symplectic forms , Math. Res. Lett. 1 (1994), 769–796

View on Pith 1994

[20] [20]

V. Turaev, Torsions of 3-dimensional manifolds, Progress in Ma thematics, vol. 208, Birkh¨ auser

View on Pith

[21] [21]

S. Wang, A vanishing theorem for Seiberg-Witten invariants , Math. Res. Lett. 2 (1995), 305–310

View on Pith 1995

[22] [22]

E. Witten, Monopoles and four-manifolds , Math. Res. Lett. 1 (1994), 769-796. Research Institute for Mathematical Sciences, Kyoto unive rsity, Kyoto, 606-8502, Japan E-mail address : nakamura@kurims.kyoto-u.ac.jp

View on Pith 1994