scholarly journals Smooth structures on nonorientable four-manifolds and free involutions

2017 ◽  
Vol 26 (13) ◽  
pp. 1750085
Author(s):  
Rafael Torres
Keyword(s):  
Existence Results ◽  
Smooth Structure ◽  
Smooth Structures ◽  
Four Manifolds

In this paper, we investigate existence of inequivalent smooth structures on closed smooth nonorientable 4-manifolds building upon results of Akbulut, Cappell–Shaneson, Fintushel–Stern, Gompf and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an application of Gluck twists to the standard smooth structure. Inspection of the smooth structure on the orientation 2-covers yields existence results of orientation-reversing exotic free involutions.

TORELLI ACTIONS AND SMOOTH STRUCTURES ON FOUR MANIFOLDS

2008 ◽  
Vol 17 (02) ◽  
pp. 171-190
Author(s):  
J. S. CALCUT
Keyword(s):  
Three Dimensional ◽  
Topological Type ◽  
Fundamental Groups ◽  
Open Book ◽  
Smooth Structure ◽  
Smooth Structures ◽  
Open Book Decompositions ◽  
Group Theoretic Method ◽  
Four Manifolds ◽  
Artin Presentation

Artin presentations are discrete equivalents of planar open book decompositions of closed, orientable three manifolds. Artin presentations characterize the fundamental groups of closed, orientable three manifolds. An Artin presentation also determines a smooth, compact, simply conected four manifold that bounds the three dimensional open book. In this way, the study of three and four manifolds may be approached purely group theoretically. In the theory of Artin presentations, elements of the Torelli subgroup act on the topology and smooth structures of the three and four manifolds. We show that the Torelli action can preserve the continuous topological type of a four manifold while changing its smooth structure. This is a new, group theoretic method of altering the smooth structure on a four manifold.

scholarly journals Smooth structures and Einstein metrics on

2009 ◽  
Vol 147 (2) ◽  
pp. 409-417 ◽  
Author(s):  
RAREŞ RǍSDEACONU ◽  
IOANA ŞUVAINA
Keyword(s):  
Scalar Curvature ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Smooth Structure ◽  
Smooth Structures ◽  
Higher Dimensional

AbstractWe show that each of the topological 4-manifolds $\bcp^2\# k\overline{\bcp^2}$, for k = 5, 6, 7, 8 admits a smooth structure which has an Einstein metric of scalar curvature s > 0, a smooth structure which carries an Einstein metric with s < 0 and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We also exhibit new examples of higher dimensional manifolds carrying Einstein metrics of both positive and negative scalar curvature.

scholarly journals Twisted Donaldson invariants

2021 ◽  
pp. 1-54
Author(s):  
TSUYOSHI KATO ◽  
HIROFUMI SASAHIRA ◽  
HANG WANG
Keyword(s):  
Gauge Theory ◽  
Fundamental Group ◽  
Picard Group ◽  
Fundamental Groups ◽  
Smooth Structure ◽  
Yang Mills ◽  
Smooth Structures ◽  
New Variant ◽  
Donaldson Invariants

Abstract Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds.

scholarly journals Spaces of knotted circles and exotic smooth structures

2020 ◽  
pp. 1-23
Author(s):  
Gregory Arone ◽  
Markus Szymik
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Homotopy Type ◽  
Free Abelian Group ◽  
Homotopy Groups ◽  
Fundamental Groups ◽  
Smooth Structure ◽  
Smooth Structures ◽  
Exotic Smooth Structures ◽  
Formula Type

Abstract Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$ have the same homotopy $(2n-7)$ -type. In the four-dimensional case, this means that the spaces of smooth knots in homeomorphic $4$ -manifolds have sets $\pi _0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi _1$ . The result about $\pi _0$ is well-known and elementary, but the result about $\pi _1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie–Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie–Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on $\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$ . We use our model to show that for every choice of basepoint, each of the homotopy groups, $\pi _1$ and $\pi _2,$ of $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ contains an infinitely generated free abelian group.

scholarly journals On normalized Ricci flow and smooth structures on four-manifolds with b+ = 1

2009 ◽  
Vol 92 (4) ◽  
pp. 355-365 ◽  
Author(s):  
Masashi Ishida ◽  
Rareş Răsdeaconu ◽  
Ioana Şuvaina
Keyword(s):  
Ricci Flow ◽  
Smooth Structures ◽  
Normalized Ricci Flow ◽  
Four Manifolds

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals Existence of Gibbs point processes with stable infinite range interaction

2020 ◽  
Vol 57 (3) ◽  
pp. 775-791
Author(s):  
David Dereudre ◽  
Thibaut Vasseur
Keyword(s):  
Point Processes ◽  
Existence Results ◽  
Pairwise Interaction ◽  
Range Interaction ◽  
Pairwise Interactions ◽  
Gibbs Point Processes ◽  
Infinite Range Interactions ◽  
Finite Configuration ◽  
Multi Body ◽  
Infinite Range

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

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