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B. Doug Park

Publications and source records attributed to B. Doug Park.

16 recordsLinked to original sources
arXiv2026

Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds

We construct infinitely many new pairwise nondiffeomorphic smooth structures on infinitely many closed simply connected spin 4-manifolds with positive signature. Our construction builds on an infinite family of simply connected complex surfaces of general type due to Roulleau and Urzúa that populate points arbitrarily near the Bogomolov-Miyaoka-Yau line in the complex geography plane. We also discuss how the conjectural symplectic Bogomolov-Miyaoka-Yau inequality implies that our results are close to being optimal.

math.GT↗
arXiv2010

Exotic smooth structures on 4-manifolds with zero signature

For every integer $k\geq 2$, we construct infinite families of mutually nondiffeomorphic irreducible smooth structures on the topological $4$-manifolds $(2k-1)(S^2\times S^2)$ and $(2k-1)(\CP#\CPb)$, the connected sums of $2k-1$ copies of $S^2\times S^2$ and $\CP#\CPb$.

math.GT↗
arXiv2010

Reverse engineering small 4-manifolds

We introduce a general procedure called `reverse engineering' that can be used to construct infinite families of smooth 4-manifolds in a given homeomorphism type. As one of the applications of this technique, we produce an infinite family of pairwise nondiffeomorphic 4-manifolds homeomorphic to CP^2#3(-CP^2).

math.GT↗
arXiv2009

Exotic Smooth Structures on Small 4-Manifolds with Odd Signatures

Let $M$ be $\CP#2\CPb$, $3\CP#4\CPb$ or $(2n-1)\CP#2n\CPb$ for any integer $n\geq 3$. We construct an irreducible symplectic 4-manifold homeomorphic to $M$ and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to $M$. We also construct such exotic smooth structures when $M$ is $\CP#4\CPb$ or $3\CP# k \CPb$ for $k=6,8,10$.

math.GT↗
arXiv2008

Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups

Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal $\checkΔ_G=\{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[\frac{n}{2}]+1$ for $A(G\times G)$. We achieve this by using the concept of the cone property in \cite{ludwig-turowska}. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed in \cite{forrest-samei-spronk}. We also present similar results for sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\times G\times G\times G$ of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets.

math.FA↗
arXiv2007

Simply connected minimal symplectic 4-manifolds with signature less than --1

For each pair $(e,σ)$ of integers satisfying $2e+3σ\ge 0$, $σ\leq -2$, and $e+σ\equiv 0\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $σ$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).

math.GT↗
arXiv2007

Constructing infinitely many smooth structures on small 4-manifolds

The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to $\CP#(2k+1)\CPb$ for $k = 1,...,4$, or to $3\CP# (2l+3)\CPb$ for $l =1,...,6$. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on $\CP#3\CPb$, $3\CP#5\CPb$ and $3\CP#7\CPb$.

math.GT↗
arXiv2006

Homologous Non-isotopic Symplectic Surfaces of Higher Genus

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.

math.GT↗
arXiv2003

Symplectic tori in rational elliptic surfaces

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface E(1) (complex projective plane blown-up at nine branch points of a generic pencil of cubic curves). We also show how these tori can be non-isotopically and symplectically embedded in many other symplectic 4-manifolds.

math.GT↗
arXiv2003

Homologous non-isotopic symplectic tori in a K3-surface

For each member of an infinite family of homology classes in the K3-surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic submanifolds in many other symplectic 4-manifolds including the elliptic surfaces E(n) for n>2.

math.GT↗
arXiv2003

Homologous Non-isotopic Symplectic Tori in Homotopy Rational Elliptic Surfaces

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive homology class in E(1)_K when K is any nontrivial fibred knot in S^3. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.

math.GT↗
arXiv2002

Non-isotopic Symplectic Tori in the Same Homology Class

For any pair of integers $n\geq 1$ and $q\geq 2$, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class $q[F]$ of an elliptic surface E(n), where $[F]$ is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedded into a more general class of symplectic 4-manifolds.

math.GT↗
arXiv2002

The Chen-Ruan Cohomology Ring of Mirror Quintic

We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror orbifold of a smooth quintic hypersurface in 4-dimensional projective space. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We outline a general method of computing the Chen-Ruan ring for Calabi-Yau hypersurfaces in projective simplicial toric varieties, modulo a conjecture that the Riemann bilinear relations are adequate for identifying the obstruction bundle for any complex orbifold.

math.AG↗