Exotic Smooth Structures Research Articles

Definite four-manifolds with exotic smooth structures

Abstract In this paper, we study smooth structures on closed, oriented four-manifolds with fundamental group Z / 2 ⁢ Z \mathbb{Z}/2\mathbb{Z} and definite intersection form. We construct infinitely many irreducible, oriented, closed, definite, smooth four-manifolds with π 1 = Z / 2 ⁢ Z \pi_{1}=\mathbb{Z}/2\mathbb{Z} and b 2 = 1 b_{2}=1 , and b 2 = 2 b_{2}=2 . As an application, we prove that, when the second Betti number b 2 b_{2} of a definite four-manifold with π 1 = Z / 2 ⁢ Z \pi_{1}=\mathbb{Z}/2\mathbb{Z} is positive and it admits a smooth structure, then it admits infinitely many smooth structures.

Evaluating TQFT invariants from $G$-crossed braided spherical fusion categories via Kirby diagrams with 3-handles

A family of TQFTs parametrised by G -crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of a G -crossed braided spherical fusion category. This reformulation is very efficient for explicit calculations, and the manifold invariant is calculated for several examples. It is also shown that in most cases, the invariant is multiplicative under connected sum, which implies that it does not detect exotic smooth structures.

From zero surgeries to candidates for exotic definite 4‐manifolds

AbstractOne strategy for distinguishing smooth structures on closed 4‐manifolds is to produce a knot in that is slice in one smooth filling of but not slice in some homeomorphic smooth filling . In this paper, we explore how 0‐surgery homeomorphisms can be used to potentially construct exotic pairs of this form. To systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find five topologically slice knots such that, if any of them were slice, we would obtain an exotic 4‐sphere. We also investigate the possibility of constructing exotic smooth structures on in a similar fashion.

Semisimple four‐dimensional topological field theories cannot detect exotic smooth structure

AbstractWe prove that semisimple four‐dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth four‐manifolds and homotopy equivalent simply connected closed oriented smooth four‐manifolds. We show that all currently known four‐dimensional field theories are semisimple, including unitary field theories, and once‐extended field theories which assign algebras or linear categories to 2‐manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4‐manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple four‐dimensional field theory is invariant under ‐stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the ‘point particles’ of the field theory. Such fermion‐free field theories cannot distinguish homotopy equivalent 4‐manifolds. Throughout, we illustrate our results with the Crane–Yetter–Kauffman field theory associated to a ribbon fusion category, settling in the negative the question of whether it is sensitive to smooth structure. As a purely algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish.

Exotic smooth structures on connected sums of S2×S2

Abstract We construct infinitely many distinct irreducible smooth structures on $n(S^2\,\times\,S^2)$, the connected sum of n copies of $S^2\,\times\,S^2$, for every odd integer $n\geq 27$.

Anomalies in (2+1)D Fermionic Topological Phases and (3+1)D Path Integral State Sums for Fermionic SPTs

Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Equivalently, our construction provides an exact (3+1)D combinatorial state sum for a path integral of any FSPT that admits a symmetry-preserving gapped boundary, including the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form $\mathbb{Z}_2$ background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the $\mathbb{Z}_{16}$ anomaly indicator for time-reversal symmetric topological superconductors with ${\bf T}^2 = (-1)^F$. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all $\mathbb{Z}_{16}$ $\mathrm{Pin}^+$ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin$^+$ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.

Particulate Exotica

Recent advances in differential topology single out four-dimensions as being special, allowing for vast varieties of exotic smoothness (differential) structures, distinguished by their handlebody decompositions, even as the coarser algebraic topology is fixed. Should the spacetime we reside in takes up one of the more exotic choices, and there is no obvious reason why it shouldn’t, apparent pathologies would inevitably plague calculus-based physical theories assuming the standard vanilla structure, due to the non-existence of a diffeomorphism and the consequent lack of a suitable portal through which to transfer the complete information regarding the exotic physical dynamics into the vanilla theories. An obvious plausible consequence of this deficiency would be the uncertainty permeating our attempted description of the microscopic world. We tentatively argue here, that a re-inspection of the key ingredients of the phenomenological particle models, from the perspective of exotica, could possibly yield interesting insights. Our short and rudimentary discussion is qualitative and speculative, because the necessary mathematical tools have only just began to be developed.

Categorical Smoothness of 4-Manifolds from Quantum Symmetries and the Information Loss Paradox.

In this paper, we focus on some aspects of the relation of spacetime and quantum mechanics and the study counterparts (in Set) of the categorical local symmetries of smooth 4-manifolds. In the set-theoretic limit, there emerge some exotic smoothness structures on (hence the Riemannian nonvanishing curvature), which fit well with the quantum mechanical lattice of projections on infinite-dimensional Hilbert spaces. The method we follow is formalization localized on the open covers of the spacetime manifold. We discuss our findings in the context of the information paradox assigned to evaporating black holes. A black hole can evaporate entirely, but the smoothness structure of spacetime will be altered and, in this way, the missing information about the initial states of matter forming the black hole will be encoded. Thus, the possible global geometric remnant of black holes in spacetime is recognized as exotic 4-smoothness. The full-fledged verification of this proposal will presumably be possible within the scope of future quantum gravity theory research.

Metrics on End-Periodic Manifolds as Models for Dark Matter

In this paper we will detail an approach to generate metrics and matter models on end-periodic manifolds, which are used extensively in the study of the exotic smooth structures of R4. After an overview of the technique, we will present two specific examples, discuss the associated matter models by solving the Einstein equations, and determine the physical viability by examining the energy conditions. We compare the resulting model directly with existing models of matter distributions in extragalactic systems, to highlight the viability of utilizing exotic smooth structures to understand the existence and distribution of dark matter.

Exotic Mazur manifolds and knot trace invariants

From a handle-theoretic perspective, the simplest contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant ν is an invariant of the trace of a knot K⊂S3, i.e. the smooth 4-manifold obtained by attaching a 2-handle to B4 along K. This provides a computable, integer-valued diffeomorphism invariant that is effective at distinguishing exotic smooth structures on knot traces and other simple 4-manifolds, including when other adjunction-type obstructions are ineffective. We also show that the concordance invariants τ and ϵ are not knot trace invariants. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct S1×S2 surgeries, resolving a question from Problem 1.16 in Kirby's list [28].

Symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg–Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically, without using any monodromy arguments. By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces.

Genus 2 Lefschetz fibrations with b 2 + = 1 and c 1 2 = 1 , 2

In this article, we construct a family of genus 2 Lefschetz fibrations f n : X θ n → S 2 with e ( X θ n ) = 11 , b 2 + ( X θ n ) = 1 , and c 1 2 ( X θ n ) = 1 by applying a single lantern substitution to the twisted fiber sums of Matsumoto’s genus 2 Lefschetz fibration over S 2 . Moreover, we compute the fundamental group of X θ n and show that it is isomorphic to the trivial group if n = − 3 or − 1 , Z if n = − 2 , and Z | n + 2 | for all integers n ≠ − 3 , − 2 , − 1 . Also, we prove that our fibrations admit − 2 section, that their total spaces are symplectically minimal, and that they have symplectic Kodaira dimension κ = 2 . In addition, using techniques developed over the past decade with other authors, we also construct the genus 2 Lefschetz fibrations over S 2 with c 1 2 = 1 , 2 and χ = 1 via the fiber sums of Matsumoto’s and Xiao’s genus 2 Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small 4 -manifolds with b 2 + = 1 and b 2 + = 3 .

Spaces of knotted circles and exotic smooth structures

AbstractSuppose 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.

Does Our Universe Prefer Exotic Smoothness?

Various experimentally verified values of physical parameters indicate that the universe evolves close to the topological phase of exotic smoothness structures on R 4 and K3 surface. The structures determine the α parameter of the Starobinski model, the number of e-folds, the spectral tilt, the scalar-to-tensor ratio and the GUT and electroweak energy scales, as topologically supported quantities. Neglecting exotic R 4 and K3 leaves these free parameters undetermined. We present general physical and mathematical reasons for such preference of exotic smoothness. It appears that the spacetime should be formed on open domains of smooth K 3 # C P 2 ¯ at extra-large scales possibly exceeding our direct observational capacities. Such potent explanatory power of the formalism is not that surprising since there exist natural physical conditions, which we state explicitly, that allow for the unique determination of a spacetime within the exotic K3.

Quantum Mechanics, Formalization and the Cosmological Constant Problem

Based on formal arguments from Zermelo–Fraenkel set theory we develop the environment for explaining and resolving certain fundamental problems in physics. By these formal tools we show that any quantum system defined by an infinite dimensional Hilbert space of states interferes with the spacetime structure M. M and the quantum system both gain additional degrees of freedom, given by models of Zermelo–Fraenkel set theory. In particular, M develops the ground state where classical gravity vanishes. Quantum mechanics distinguishes set-theoretic random forcing such that M and gravitational degrees of freedom are parameterized by extended real line. The large scale smooth geometry compatible with the forcing extensions is one of exotic smoothness structures of $${\mathbb {R}}^4$$ . The amoeba forcing makes the old real line to have Lebesgue measure zero in the extended one. We apply the entire procedure to the cosmological constant problem, especially to discard the zero-modes contributions to the gravitational vacuum density. Moreover, there exists certain exotic smooth $${\mathbb {R}}^4$$ from which one determines the realistic, agreeing with observation, small value of the vacuum energy density.

Torus solutions to the Weierstrass-Enneper representation of surfaces

In this paper, we present a torus solution to the generalized Weierstrass-Enneper representation of surfaces in R4. The key analytical technique will be Bloch wave functions with complex wave vectors. We will also discuss some possible uses of these solutions which derive from their explicit nature, such as Dehn surgery and the physics of exotic smooth structure.

A topological approach to neutrino masses by using exotic smoothness

In this paper, we will consider a cosmological model with two topological transitions of the space. The smooth 4-dimensional spacetime of the model admits topological transitions of its 3-dimensional slices. The whole approach is inspired by a class of exotic smoothness structures on [Formula: see text]. In particular, this class of smoothness structures induces two topological transitions. Then, we are able to calculate the energy scales as associated to these topological transitions. For the first transition we will get the value of the GUT scale and the energy of the second transition is at the electroweak scale. The topology of the exotic [Formula: see text] determines both, the energy of the scales by certain topological invariants, and the existence of the right-handed sterile neutrino. It is the input for the seesaw mechanism. Secondly, based on this model, we are able to calculate the neutrino masses which are in a very good agreement with experiments. Finally, we will speculate, again based on topology, why there are three generations of neutrinos and an asymmetry between neutrinos and anti-neutrinos.

Trisections of 4-manifolds

The study of n-dimensional manifolds has seen great advances in the last half century. In dimensions greater than four, surgery theory has reduced classification to homotopy theory except when the fundamental group is nontrivial, where serious algebraic issues remain. In dimension 3, the proof by Perelman of Thurston’s Geometrization Conjecture (1) allows an algorithmic classification of 3-manifolds. The work of Freedman (2) classifies topological 4-manifolds if the fundamental group is not too large. Also, gauge theory in the hands of Donaldson (3) has provided invariants leading to proofs that some topological 4-manifolds have no smooth structure, that many compact 4-manifolds have countably many smooth structures, and that many noncompact 4-manifolds, in particular 4D Euclidean space R 4 , have uncountably many. However, the gauge theory invariants run into trouble with small 4-manifolds, such as those with the same homology groups as the 4D sphere, S 4 . In particular, the smooth 4D Poincare Conjecture, the last remaining case of that hallowed conjecture, is still open. (In higher dimensions, the smooth Poincare Conjecture is sometimes true in the following sense. In dimensions 3, 5, 6, 12, and 61, a homotopy sphere is diffeomorphic to the standard one, and in all other known cases, there are increasingly many exotic smooth structures on the topological sphere; however, it is possible that there may be more high-dimensional cases with no exotic spheres.) The gauge theory invariants are very good at distinguishing smooth 4-manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic. What is missing is the equivalent of the higher-dimensional s-cobordism theorem, a key to the successes in higher dimensions. The s-cobordism theorem states that, if M 0 m and M 1 m are the two boundary components of an m + 1 -dimensional manifold W and if … [↵][1]1Email: kirby{at}math.berkeley.edu. [1]: #xref-corresp-1-1

Exotic open 4–manifolds which are nonleaves

We study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open $4$-manifolds which are not diffeomorphic to any leaf of a codimension one transversely $C^{2}$ foliation on a compact manifold. These examples include some exotic ${\mathbb R}^4$'s and exotic cylinders $S^3\times{\mathbb R}$.

Nonexistence of Stein structures on 4-manifolds and maximal Thurston–Bennequin numbers

For a 4-manifold represented by a framed knot in $S^3$, it has been well known that the 4-manifold admits a Stein structure if the framing is less than the maximal Thurston-Bennequin number of the knot. In this paper, we prove either the converse of this fact is false or there exists a compact contractible oriented smooth 4-manifold (with Stein fillable boundary) admitting no Stein structure. Note that an exotic smooth structure on $S^4$ exists if and only if there exists a compact contractible oriented smooth 4-manifold with $S^3$ boundary admitting no Stein structure.