Monograph compares curve signs in intrinsic and Pin-dependent definitions.
problem Comparing curve signs in intrinsic and Pin-dependent definitions.
method Intrinsic perspective on Spin- and Pin-structures, detailed constructions of orientations, explicit comparison.
result Explicit comparison between curve signs in intrinsic and Pin-dependent definitions.
The paper explores properties of Pin structures on surfaces and their cobordism.
problem Understanding Pin structures on compact surfaces and their cobordism.
method Analyzes Pin structures on surfaces and their cobordism, providing a construction for dimensions up to two.
result Pin structures on surfaces differ by a diffeomorphism if and only if they are cobordant, but this does not extend to higher dimensions.
The paper studies extPin±-structures on non-oriented 4-manifolds via Lefschetz fibrations.
problem Understanding extPin±-structures on non-oriented 4-manifolds and Lefschetz fibrations. method Extending work on orientable settings, the paper uses Lefschetz fibrations to analyze extPin±-structures on non-orientable 4-manifolds and vector bundles. result The paper provides existence results of extPin+ and extPin−-structures on closed non-orientable 4-manifolds and Lefschetz fibrations over the 2-sphere. This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement of the mod 2 intersection form, q: H_1(F;Z/2Z) -> Z/4Z Theorem 1.1 says that q v…
Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.
problem Understanding real lines on real del Pezzo surfaces of degree 1.
method Intrinsically defined Pin-structure on the real locus.
result The difference between hyperbolic and elliptic lines is always 16.
A correspondence between different Pin-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.
Geometrically interprets cup products and defines combinatorial Pin structures.
problem Understanding Steenrod's cup products and their geometric interpretation.
method Constructs vector fields and combinatorial frames to interpret cochain-level formulas.
result Geometrically interprets cup products and defines Pin structures combinatorially.
The paper classifies surfaces in 4-manifolds up to concordance.
problem Classifying surfaces in 4-manifolds up to concordance.
method Ambient surgery with Pin−-structures, cobordism theory, and topological constructions. result Closed surfaces in simply-connected 4-manifolds are concordant if cobordant.
The main result of this paper is a Pfaffian formula for the partition function of the dimer model on a graph G embedded in a closed, possibly non-orientable surface S. This formula is suitable for computational purposes, and it is obtained using purely geometrical methods. The key step in the proof consists of a corres…
We show that if M is an orientable 4-dimensional infrasolvmanifold and either β=β1(M;Q)≥2 or M is a Sol04- or a Solm,n4-manifold (with m=n) then M is parallelizable. There are non-parallelizable examples with β=1 for each of the other solvable Lie geometries $\ma…
Generalizes Rochlin's theorem to 4-manifolds with boundary and arbitrary 3-manifold boundaries.
problem Proving divisibility of self-intersection numbers in 4-manifolds with boundary.
method Combining Kirby-Siebenmann invariants, Brown invariants, and pin structures.
result Combinatorial description of Kirby-Siebenmann invariant for 4-manifolds with boundary.
We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure…
New classification for some unorientable 4-manifolds using modified surgery theory.
problem Classifying stable diffeomorphism classes of unorientable 4-manifolds.
method Modified surgery theory applied to unorientable 4-manifolds with specific fundamental groups.
result Found nine stable diffeomorphism classes for pin+ manifolds, one for pin−, and four for neither, under certain conditions. Generalizes JT gravity to time-reversal symmetric theories with fermions and supersymmetry.
problem Matching JT gravity and random matrix ensembles with different symmetries.
method Extends techniques from hermitian matrices to other ensembles, including super Riemann surfaces.
result Volume of moduli spaces in JT gravity and supergravity is related to random matrix volumes.