arXiv research

A new method shrinks a complex structure without much change.

New bounds on the wildness of Bing's involution.

We show that, for each $1\le p < 2$, there exists a wild involution $\mathbb S^3\to \mathbb S^3$ in the Sobolev class $W^{1,p}(\mathbb S^3,\mathbb S^3)$.

Bing's house-like spines approximate all PL manifolds.

Bing doubling is an operation which gives a satellite of a knot. It is also applied to a link by specifying a component of the link. We give a formula to compute the reduced colored Jones polynomial of a Bing double by using that of the companion. This formula enables us to compute a lot of examples of the reduced colo…

We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for n>1 the (n+1)-st iterated Bing double of a knot is rationally slice if and only if the n-th iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying …

Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger…

We construct Bing houses in all dimensions $n \geq 3$, obtaining non-separating PL immersions of $\mathbb{S}^n \rightarrow\mathbb{R}^{n+1}$.

Bing doubling is an operation which produces a 2-component boundary link B(K) from a knot K. If K is slice, then B(K) is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if B(K) is boundary slice, then K is algebraically slice. We also show that the Ras…

Cha and Kim proved that if a knot K is not algebraically slice, then no iterated Bing double of K is concordant to the unlink. We prove that if K has nontrivial signature $σ$, then the n-iterated Bing double of K is not concordant to any boundary link with boundary surfaces of genus less than $2^{n-1}σ$. The same resul…

We show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive…

If the Bing double of a knot K is slice, then K is algebraically slice. In addition, Heegaard--Floer concordance invariants developed by Ozsvath-Szabo and by Manolescu-Owens vanish on K.

We give a short proof of Bing's characterization of $S^3$: a compact, connected 3-manifold $M$ is $S^3$ if and only if every knot in $M$ is isotopic into a ball.

Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\colon X\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$ with the source limitati…

The notion of a Bing cell is introduced, and it is used to define invariants, link groups, of 4-manifolds. Bing cells combine some features of both surfaces and 4-dimensional handlebodies, and the link group λ(M) measures certain aspects of the handle structure of a 4-manifold M. This group is a quotient of the fundame…

The paper explores coalescent contractions in contractible spaces, providing criteria and examples.

Let K be a knot in S^3. We study the iterated Bing doubles of K, giving a new proof for the following statement: If BD_n(K) is slice for some n, then K is algebraically slice. This result was first proved by Cha and Kim using covering link calculus. We also use this tool, but our proof is substantially simpler and illu…

We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every $n$-dimensional homogeneous ANR is a topological $n$-manifold, whereas the Busemann Conjecture asserts that every $n$-dimensional $G$-space is a topological $n$-manifold. The key object in bo…

In accordance with the Bing-Borsuk conjecture \cite{bb}, we show that if $X$ is an $n$-dimensional homogeneous metric $ANR$ compactum and $x\in X$, then there is a local basis at x consisting of connected open sets U such that the homological properties of \bar U and bdU are similar to the properties of the closed ball…

In this paper, we propose an offline counterfactual policy estimation framework called Genie to optimize Sponsored Search Marketplace. Genie employs an open box simulation engine with click calibration model to compute the KPI impact of any modification to the system. From the experimental results on Bing traffic, we s…

We introduce a technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner.…

A quick proof of Bing's theorem indicated by the title is given. The proof also concludes Gumerov's result on covering degrees of solenoids.

The abstract discusses knots and their isotopy to the unknot, proving a specific case.

Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold Y can be embedded in M so that the exterior of the imag…

The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.

We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements …

Nonpositive towers property in 3-manifolds spines.

We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion wh…

Classifies involutions on spherical 3-manifolds.

Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.

Perelman's proof confirmed, new method uses 4D topology.

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

New formula for dual knots using involutions.

The study classifies involutions on del Pezzo surfaces.

Characterizes the Legendre involution on generic frontals.

The paper defines conditions for good involutions in generalized Alexander quandles.

Study proves naturality and functoriality in a type of Heegaard Floer homology.

The paper develops a new theory for knots and 3-manifolds with involutions.

The study proves symplectic quandles cannot have good involutions.

Involutions generate mapping class groups of infinite surfaces.

Minimal involutions generate a subgroup of nonorientable surfaces.

Study exact surgery formula in involutive Heegaard Floer homology.

We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \geqslant 3$ PU($n$,1) has involution length at most 8.

Anti-symplectic involutions connect a sphere in a symplectic surface.

Real slices of parabolic opers on Riemann surfaces are studied.

Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.

Computed involutive knot invariants for specific pretzel knots.

Presentations for involutions on non-orientable surfaces up to genus 5.