Exotic Smooth Structures Research Articles

Smooth structures, normalized Ricci flows, and finite cyclic groups

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group \({\mathbb{Z}_{d}}\) , where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group \({\mathbb{Z}_{d}}\) , which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular \({\mathbb{Z}_{d}}\) -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of \({\mathbb{CP}^{2}\rm{s}}\) and \({\overline{\mathbb {CP}}^{2}\rm{s}}\).

Symmetries and exotic smooth structures on a K3 surface

Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and a comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and, as a result, the nonsmoothability of actions induced by a holomorphic automorphism of prime order at least 7 is proved, and the nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 that has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of prime order at least 5 is explicitly determined, provided that the action is homologically nontrivial.

Exotic smooth structures on small 4-manifolds

Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first example of a simply-connected symplectic 4-manifold that is homeomorphic but not diffeomorphic to M.

Surgery on nullhomologous tori and simply connected 4-manifolds withb+= 1

For 5 ⩽ k ⩽ 8, we show that an infinite family of exotic smooth structures on CP 2 # k CP ¯ 2 can be obtained by 1/n-surgeries on a single embedded nullhomologous torus in a manifold Rk which is homeomorphic to CP2#k CP 2 # k CP ¯ 2 .

Exotic smooth structures and symplectic forms on closed manifolds

In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.

Four-dimensional Manifolds

Four-dimensional Manifolds

Exotic smooth structures on 3CP2♯8CP¯2

It is an intriguing and very tricky problem in differential topology to construct exotic smooth 4-manifolds with small Euler characteristics. For example, given an ordered pair of positive integers (i, n), we can ask whether there exists a smooth closed simply-connected 4-manifold that is homeomorphic but not diffeomorphic to mCP2#nCP . Note that such an exotic 4-manifold has e = m + n + 2, b+ = m, and b= n. Probably the most famous example of this kind is the Barlow surface, which corresponds to the pair (1, 8) (cf. [B] and [Ko]). The Barlow surface has the smallest Euler characteristic among all known closed oriented simply-connected 4manifolds with more than one smooth structure. In this paper we construct three exotic 4-manifolds corresponding to the pairs (3,10) and (3,12). Namely, we prove the following

Construction of exotic smooth structures

In this article, we construct infinitely many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds starting from the elliptic surfaces E ( n ) and applying the sequence of knot surgery, ordinary blowups and rational blowdown. We also compute the Seiberg–Witten invariants of these manifolds.

A note on logarithmic transformations on the Hopf surface

In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on 4-manifolds. We will show here that this is not the case for the Hopf surface, all integer homology Hopf surfaces we obtain are diffeomorphic to the standard Hopf surface. To cite this article: R. Zentner, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Simply connected symplectic 4-manifolds with b2+=1 and c12=2

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface ${\mathbf CP}^2 \sharp 7{\bar{{\mathbf CP}}^2}$ admits an exotic smooth structure.

Exotic Smoothness and Noncommutative Spaces. The Model-Theoretical Approach

We give an almost explicit presentation of exotic functions corresponding to some exotic smooth structure on topologically trivial ℝ4. The construction relies on the model-theoretic tools from the previous paper. We can formulate unexpected, yet direct connection between “localized” exotic small R4's and some noncommutative spaces. The formalism of QM can be interpreted in terms of exotic smooth R4's localized in spacetime. A new way of looking at the problem of decoherence is suggested. The 4-dimensional spacetime itself has built-in means which may enforce a kind of decoherence.

Background Independence in Quantum Gravity and Forcing Constructions

A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ℝ4. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory.

Exotic smooth structures on 4-manifolds, II

In this article we continue to investigate exotic smooth structures of 4-manifolds studied in [Forum Math. 14 (2002) 915–929]. As a conclusion, we claim that most known simply connected, closed, irreducible, nonspin, smooth 4-manifolds with b 2 + odd and large enough admit infinitely many, both symplectic and non-symplectic, exotic smooth structures.

Set Theoretical Forcing in Quantum Mechanics and AdS/CFT Correspondence

We show unexpected connection of Set Theoretical Forcing with Quantum Mechanical lattice of projections over some separable Hilbert space. The basic ingredient of the construction is the rule of indistinguishability of Standard and some Nonstandard models of Peano Arithmetic. The ingeneric reals introduced by M. Ozawa will correspond to simultaneous measurement of incompatible observables. We also discuss some results concerning model theoretical analysis of Small Exotic Smooth Structures on topological 4-space ℝ4. Forcing appears rather naturally in this context and the rule of indistinguishability is crucial again. As an unexpected application we are able to approach Maldacena Conjecture on AdS/CFT correspondence in the case of AdS 5 × S 5 and Super YM Conformal Field Theory in four dimensions. We conjecture that there is possibility of breaking Supersymetry via sources of gravity generated in four dimensions by exotic smooth structures on ℝ4 emerging in this context.

The geography of spin symplectic 4-manifolds

In this paper we construct a family of simply connected spin non-complex symplectic 4-manifolds which cover all but finitely many allowed lattice points (\(\chi,{\mathbf c}\)) lying in the region \(0 \leq{\mathbf c} \leq 8.76\chi\). Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on \((2n+1)(S^2\times S^2)\) for all n large enough.

Exotic smooth structures on nonpositively curved symmetric spaces

We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the change of the smooth structure we use a tangential map from the locally symmetric space its dual compact type twin.

Exotic smooth structures on 4-manifolds

Exotic smooth structures on 4-manifolds

Exotic involutions on symplectic 4-manifolds and G-monopole invariants

Using the G-monopole invariant, we shall show that any anti-holomorphic involution on a closed symplectic 4-manifold is not diffeomorphic to any holomorphic involution. As a corollary, we shall give a way to construct exotic smooth structures.

Complex hyperbolic manifolds and exotic smooth structures

Complex hyperbolic manifolds and exotic smooth structures

KNOTS AND EXOTIC SMOOTH STRUCTURES ON 4-MANIFOLDS

In this paper we give a short survey of exotic smooth structures of 4-manifolds. Various methods of constructing exotic structures and their relation to “Shake-Sliceness” and the Zeeman conjecture is discussed.