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**Mod \(p\) vanishing theorem of Seiberg-Witten invariants for 4-manifolds with \(\mathbb Z_ p\)-actions.**
*(English)*
Zbl 1129.57038

The paper under review is to give an alternative proof of the mod \(p\) vanishing result by F. Fang [Int. J. Math. 9, No. 8, 957–973 (1998; Zbl 0922.57013)] of Seiberg-Witten invariants under a cyclic group action of prime order, and to extend the result to the case \(b_1 \geq 1\) in Theorem 1.2 of the paper.

In section 2, the author reviews the \(G\)-equivariant finite dimensional approximation introduced by Furuta. The \(G\)-equivariant perturbation of the Seiberg-Witten monopole map achieves transversality and the zero set of the perturbed monopole moduli space is fixed-point free as shown in section 3.1 for dimensional reasons. The author proves his main result when the dimension of the Seiberg-Witten moduli space is zero in section 3.2, and when the dimension is positive and even in section 3.3, and the rest in section 3.4 by a geometric method which is different from Fang’s approach.

For \(b_1 \geq 1\), the whole theory is viewed as a family on the Jacobian torus \(J\), its fixed point set \(J^G\) is decomposed into connected components. By restriction, there is a homomorphism \(r_L: K_G(J) \to K_G(t_l)\) for a point \(t_l \in J_l \subset J^G\). The Seiberg-Witten invariants from the tori in the Jacobian are studied in section 4 to cut down the dimension from the monopole moduli spaces. The author gives several examples when \(G=\mathbb Z_2\) in section 5.2 and when \(G=\mathbb Z_3\) in section 5.3 for the main theorem 1.2 in the paper.

It would be interesting to know what the corresponding result would be if the order of the action group is not prime (for example \(|G| = pq\) for \((p, q) =1\)).

In section 2, the author reviews the \(G\)-equivariant finite dimensional approximation introduced by Furuta. The \(G\)-equivariant perturbation of the Seiberg-Witten monopole map achieves transversality and the zero set of the perturbed monopole moduli space is fixed-point free as shown in section 3.1 for dimensional reasons. The author proves his main result when the dimension of the Seiberg-Witten moduli space is zero in section 3.2, and when the dimension is positive and even in section 3.3, and the rest in section 3.4 by a geometric method which is different from Fang’s approach.

For \(b_1 \geq 1\), the whole theory is viewed as a family on the Jacobian torus \(J\), its fixed point set \(J^G\) is decomposed into connected components. By restriction, there is a homomorphism \(r_L: K_G(J) \to K_G(t_l)\) for a point \(t_l \in J_l \subset J^G\). The Seiberg-Witten invariants from the tori in the Jacobian are studied in section 4 to cut down the dimension from the monopole moduli spaces. The author gives several examples when \(G=\mathbb Z_2\) in section 5.2 and when \(G=\mathbb Z_3\) in section 5.3 for the main theorem 1.2 in the paper.

It would be interesting to know what the corresponding result would be if the order of the action group is not prime (for example \(|G| = pq\) for \((p, q) =1\)).

Reviewer: Weiping Li (Stillwater)

### MSC:

57R57 | Applications of global analysis to structures on manifolds |

57S17 | Finite transformation groups |

57M60 | Group actions on manifolds and cell complexes in low dimensions |