Circle actions on simply connected $4$-manifolds

Fintushel, Ronald

doi:10.1090/s0002-9947-1977-0458456-6

Cited by 56 publications

(139 citation statements)

References 9 publications

Supporting

Mentioning

139

Contrasting

Order By: Relevance

“…Identify SU (2) with the unit quaternions. The map Ad : SU (2) → SO(3) is given by g → ghḡ for all h ∈ ImH and is the double cover of SO (3). Thus SO(3) can be thought of as the unit quaternions modulo the equivalence h ∼ −h.…”

Section: Theoremmentioning

confidence: 99%

Seiberg-Witten invariants, orbifolds, and circle actions

Baldridge

2002

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The main result of this paper is a formula for calculating the Seiberg-Witten invariants of 4-manifolds with fixed-point-free circle actions. This is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4-manifold and the moduli space of the quotient 3-orbifold. Two corollaries include the fact that b + >1 4-manifolds with fixed-point-free circle actions are simple type and a new proof of the equality SW Y 3 ×S 1 = SW Y 3 . An infinite number of 4-manifolds with b + = 1 whose Seiberg-Witten invariants are still diffeomorphism invariants is constructed and studied.

show abstract

Section: Theoremmentioning

confidence: 99%

Seiberg-Witten invariants, orbifolds, and circle actions

Baldridge

2002

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Fintushel used this weighted space to distinguish simply connected 4-manifolds with circle actions [9], [10]. We will need this result also.…”

Section: Proofsmentioning

confidence: 99%

Seiberg–Witten vanishing theorem for S¹-manifolds with fixed points

Baldridge¹

2004

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…By a method described in [Fin1], [Fin2] we can introduce a local T 2 action extending the T 1 action in a neighbourhood of the set lying above Fix…”

Section: Proof Let Us Observe That Mmentioning

confidence: 99%

“…Using the argument from [Fin1] and [Fin2] or a description of normal bundle to singular strata in [D] we can further extend the T 2 action to a local T 2 action on small tubular neighbourhood of the closure of the set of points with finite isotropy subgroups (including the intervals with Z 1 isotropy subgroup).…”

Section: Proof Let Us Observe That Mmentioning

confidence: 99%

Bordism of spin 4-manifolds with local action of tori

Mikrut

1998

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We prove that bordism group of spin 4-manifolds with singular Tstructure, the notion introduced by Cheeger and Gromov, is an infinite cyclic group and is detected by singnature. In particular we obtain that the theorem of Atiyah and Hirzebruch of vanishing ofÂ-genus in case of S 1 action on spin 4n-manifolds is not valid in case of T -structures on spin 4-manifolds. IntroductionThe notion of a local action of tori on a manifold is a generalization of an action of a torus. Tori, possibly of different dimensions, act effectively on open subsets of the manifold and these actions fit together on overlaps in such a way that the torus acting on one of the sets injects homomorphically into the torus acting on the second one. If we assume that each of these actions is without fixed points, then the local action of tori coincides with a T -structure. The notion of a T -structure and more general notions of an F structure and a nilpotent Killing structure appeared [CFG] in the context of collapsing Riemannian manifolds.Here we assume that the manifold is compact, differentiable, spin and that the local action of tori may admit fixed points. The main result of the paper is Theorem. The bordism group of compact spin 4-manifolds admitting local actions of tori is isomorphic to Z. There is a generator with signature 16.In particular we obtain that any compact spin 4-manifold is spin cobordant with a manifold admitting local action of tori. In dimension 4 Sign(M ) = − According to the present knowledge of the author the statement of the conjecture is true for pure local action of tori on 4-manifolds and under certain assumptions on the dimension of the acting torus for 8-manifolds.

show abstract

Circle actions on simply connected $4$-manifolds

Cited by 56 publications

References 9 publications

Seiberg-Witten invariants, orbifolds, and circle actions

Seiberg-Witten invariants, orbifolds, and circle actions

Seiberg–Witten vanishing theorem for S¹-manifolds with fixed points

Bordism of spin 4-manifolds with local action of tori

Contact Info

Product

Resources

About

Circle actions on simply connected $4$-manifolds

Cited by 56 publications

References 9 publications

Seiberg-Witten invariants, orbifolds, and circle actions

Seiberg-Witten invariants, orbifolds, and circle actions

Seiberg–Witten vanishing theorem for S1-manifolds with fixed points

Bordism of spin 4-manifolds with local action of tori

Contact Info

Product

Resources

About

Seiberg–Witten vanishing theorem for S¹-manifolds with fixed points