Smooth structures on nonorientable four-manifolds and free involutions

Torres, Rafael

doi:10.1142/s0218216517500857

Cited by 4 publications

(3 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Whether this is possible in the orientable setting remains open. Other examples of the operation changing the smooth structure on nonorientable 4-manifolds are given in [135], [71,Proposition 1.6]. See also [10] for a condition that implies the Gluck twist operation does not change the diffeomorphism type.…”

Section: Teichner's E # E and E # Ementioning

confidence: 99%

Counterexamples in 4-manifold topology

Kasprowski

Powell

Ray

2023

EMS Surv. Math. Sci.

View full text Add to dashboard Cite

We illustrate the rich landscape of 4-manifold topology through the lens of counterexamples. We consider several of the most commonly studied equivalence relations on 4-manifolds and how they are related to one another. We explain implications e.g. that h-cobordant manifolds are stably homeomorphic, and we provide examples illustrating the failure of other potential implications. The information is conveniently organised in a flowchart and a table.

show abstract

Section: Teichner's E # E and E # Ementioning

confidence: 99%

Counterexamples in 4-manifold topology

Kasprowski

Powell

Ray

2023

EMS Surv. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…Whether this is possible in the orientable setting remains open. Other examples of the operation changing the smooth structure on nonorientable 4-manifolds are given in [Tor17]. See also [AY13] for a condition that implies the Gluck twist operation does not change the diffeomorphism type.…”

Section: • By Combining With Their Earlier Work On 4-dimensional Surg...mentioning

confidence: 99%

Counterexamples in 4-manifold topology

Kasprowski¹,

Powell²,

Ray³

2022

Preprint

View full text Add to dashboard Cite

We consider several of the most commonly studied equivalence relations on 4manifolds and how they are related to one another. We explain implications e.g. that h-cobordant manifolds are stably homeomorphic, and we provide examples illustrating the failure of other potential implications. The information is conveniently organised in a flowchart and a table. 2020 Mathematics Subject Classification. 57K40.

show abstract

“…That is Q := P S is homeomorphic [Wan95] but not diffeomorphic to P . Torres [Tor17] extended this construction to other nonorientable 4-manifolds.…”

Section: Introductionmentioning

confidence: 99%

Gluck twists on concordant or homotopic spheres

Kasprowski¹,

Powell²,

Ray³

2022

Preprint

View full text Add to dashboard Cite

Let M be a compact 4-manifold and let S and T be embedded 2-spheres in M , both with trivial normal bundle. We write M S and M T for the 4-manifolds obtained by the Gluck twist operation on M along S and T respectively. We show that if S and T are concordant, then M S and M T are s-cobordant, and so if π 1 (M ) is good, then M S and M T are homeomorphic. Similarly, if S and T are homotopic then we show that M S and M T are simple homotopy equivalent. Under some further assumptions, we deduce that M S and M T are homeomorphic. We show that additional assumptions are necessary by giving an example where S and T are homotopic but M S and M T are not homeomorphic. We also give an example where S and T are homotopic and M S and M T are homeomorphic but not diffeomorphic.

show abstract

Smooth structures on nonorientable four-manifolds and free involutions

Cited by 4 publications

References 28 publications

Counterexamples in 4-manifold topology

Counterexamples in 4-manifold topology

Counterexamples in 4-manifold topology

Gluck twists on concordant or homotopic spheres

Contact Info

Product

Resources

About