Mark Powell

Linking forms of amphichiral knots

Stefan Friedl, Allison N. Miller, Mark Powell

We give a simple obstruction for a knot to be amphichiral, in terms of the homology of the 2-fold branched cover. We work with unoriented knots, and so obstruct both positive and n…

Pseudo-isotopies of simply connected 4-manifolds

David Gabai, David T. Gay, Daniel Hartman +2

Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another re…

Cosmetic crossings and Seifert matrices

Cheryl Balm, Stefan Friedl, Efstratia Kalfagianni +1

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that…

Stable classification of 4-manifolds with 3-manifold fundamental groups

Daniel Kasprowski, Markus Land, Mark Powell +1

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if an…

math.AT2026

Simple homotopy types of even dimensional manifolds

Csaba Nagy, John Nicholson, Mark Powell

Given a closed $n$ -manifold, we consider the set of simple homotopy types of $n$ -manifolds within its homotopy type, called its simple homotopy manifold set. We characterise it in…

The 4-dimensional disc embedding theorem and dual spheres

Mark Powell, Arunima Ray, Peter Teichner

We modify the proof of the disc embedding theorem for $4$ -manifolds, which appeared as Theorem 5.1A in the book "Topology of 4-manifolds" by Freedman and Quinn, in order to constru…

Whitney towers and abelian invariants of knots

Jae Choon Cha, Kent E. Orr, Mark Powell

We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball…

math.GT2015

Smoothly slice boundary links whose derivative links have nonvanishing Milnor invariants

Hye Jin Jang, Min Hoon Kim, Mark Powell

We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert f…

Embedded surfaces with infinite cyclic knot group

Anthony Conway, Mark Powell

We study locally flat, compact, oriented surfaces in $4$ -manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfa…

Counterexamples in 4-manifold topology

Daniel Kasprowski, Mark Powell, Arunima Ray

We illustrate the rich landscape of 4-manifold topology through the lens of counterexamples. We consider several of the most commonly studied equivalence relations on 4-manifolds a…

math.GT2015

Casson towers and slice links

Jae Choon Cha, Mark Powell

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which a…

A survey of the foundations of four-manifold theory in the topological category

Stefan Friedl, Matthias Nagel, Patrick Orson +1

This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise r…

Mapping class groups of simply connected 4-manifolds with boundary

Patrick Orson, Mark Powell

We compute the topological mapping class group of every compact, simply connected, topological 4-manifold. This was previously only known in the closed case. If the 4-manifold is s…

Characterisation of homotopy ribbon discs

Anthony Conway, Mark Powell

Let $I^$ be either the infinite cyclic group $Z$ or the Baumslag-Solitar group $Z ⋉ Z [\frac{1}{2}]$ . Let $K$ be a slice knot admitting a slice disc…

Homotopy classification of $4$ -manifolds with $3$ -manifold fundamental group

Jonathan Hillman, Daniel Kasprowski, Mark Powell +1

We give a criterion on a group $I ¨ $ and a homomorphism $w : I ¨ \to C_{2}$ under which closed $4$ -manifolds with fundamental group $I ¨ $ and orientation character $w$ are classifie…

math.GT2016

Twisted Blanchfield pairings and symmetric chain complexes

Mark Powell

We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involu…

Satellites and concordance of knots in 3-manifolds

Stefan Friedl, Matthias Nagel, Patrick Orson +1

Given a 3-manifold $Y$ and a free homotopy class in $[S^{1}, Y]$ , we investigate the set of topological concordance classes of knots in $Y \times [0, 1]$ representing the given homotop…

Simply-connected manifolds with large homotopy stable classes

Anthony Conway, Diarmuid Crowley, Mark Powell +1

For every $k \geq 2$ and $n \geq 2$ we construct $n$ pairwise homotopically inequivalent simply-connected, closed $4 k$ -dimensional manifolds, all of which are stably diffeomorphic…

math.GT2022

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

Allison N. Miller, Mark Powell

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertib…

Simple spines of homotopy 2-spheres are unique

Patrick Orson, Mark Powell

A locally flatly embedded $2$ -sphere in a compact $4$ -manifold $X$ is called a spine if the inclusion map is a homotopy equivalence. A spine is called simple if the complement of t…

Stably exotic 4-manifolds

Daniel Kasprowski, Mark Powell

A pair of closed, smooth $4$ -manifolds $M$ and $M^{'}$ are stably exotic if they are stably homeomorphic but not stably diffeomorphic, where stabilisation refers to connected sum with…

Two-solvable and two-bipolar knots with large four-genera

Jae Choon Cha, Allison N. Miller, Mark Powell

For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that 2-solvable knots are in particular algebraically slice and…

Spanning 3-discs in the 4-sphere pushed into the 5-disc

Mark Powell

I prove that any two smooth collections of spanning 3-discs for the trivial 2-link in $S^{4}$ become smoothly isotopic rel. boundary after pushing them into $D^{5}$ .

Twisted Blanchfield pairings and decompositions of 3-manifolds

Stefan Friedl, Constance Leidy, Matthias Nagel +1

We prove a decomposition formula for twisted Blanchfield pairings of 3-manifolds. As an application we show that the twisted Blanchfield pairing of a 3-manifold obtained from a 3-m…

Gluck twists on concordant or homotopic spheres

Daniel Kasprowski, Mark Powell, Arunima Ray

Let $M$ be a compact $4$ -manifold and let $S$ and $T$ be embedded $2$ -spheres in $M$ , both with trivial normal bundle. We write $M_{S}$ and $M_{T}$ for the $4$ -manifolds obtained by th…

A lower bound for the doubly slice genus from signatures

Patrick Orson, Mark Powell

The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the c…

A ribbon obstruction and derivatives of knots

JungHwan Park, Mark Powell

We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative link…

Triple linking numbers and surface systems

Christopher William Davis, Matthias Nagel, Patrick Orson +1

We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. Given two links with the same pairwise linking numbers we show that they have t…

math.GT2016

A calculation of Blanchfield pairings of 3-manifolds and knots

Stefan Friedl, Mark Powell

We calculate Blanchfield pairings of 3-manifolds. In particular, we give a formula for the Blanchfield pairing of a fibred 3-manifold and we give a new proof that the Blanchfield p…

cs.LG2026

Closing Gaps in Emissions Monitoring with Climate TRACE

Brittany V. Lancellotti, Jordan M. Malof, Aaron Davitt +32

Global greenhouse gas emissions estimates are essential for monitoring and mitigation planning. Existing emissions datasets provide critical foundations for understanding emissions…

Links Not Concordant to the Hopf Link

Stefan Friedl, Mark Powell

We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.

Link concordance implies link homotopy

Maciej Borodzik, Mark Powell, Peter Teichner

We show that link concordance implies link homotopy for immersions of codimension at least two. As a consequence, we prove that every link $⊔^{r} S^{n} ↪ S^{n + 2}$ is…

Doubly slice knots and metabelian obstructions

Patrick Orson, Mark Powell

For $ℓ > 1$ , we develop $L^{(2)}$ -signature obstructions for $(4 ℓ - 3)$ -dimensional knots with metabelian knot groups to be doubly slice. For each $ℓ > 1$ , we construct an infi…

Smooth stable isotopy of topologically isotopic surfaces

Daniel Galvin, Patrick Orson, Mark Powell

A stabilisation of a $4$ -manifold $X$ is the connected sum of $X$ with some number of copies of $S^{2} \times S^{2}$ . If two smooth surfaces in a $4$ -manifold are topologically isotopic…

Corks for exotic diffeomorphisms

Vyacheslav Krushkal, Anubhav Mukherjee, Mark Powell +1

We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabil…

Smoothly slice knots with smoothly non-approximable topological slice discs

Min Hoon Kim, Mark Powell

We construct infinitely many smoothly slice knots having topological slice discs that are non-approximable by smooth slice discs.

Examples of topologically unknotted tori

AndrÃ¡s JuhÃ¡sz, Mark Powell

We show that certain smooth tori with group $Z$ in $S^{4}$ have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the…

math.GT2013

Embedded Morse theory and relative splitting of cobordisms of manifolds

Maciej Borodzik, Mark Powell

We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding i…

math.GT2013

Splitting numbers of links

Jae Choon Cha, Stefan Friedl, Mark Powell

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new tech…

$4$ -manifolds with boundary and fundamental group $Z$

Anthony Conway, Lisa Piccirillo, Mark Powell

We classify topological $4$ -manifolds with boundary and fundamental group $Z$ , under some assumptions on the boundary. We apply this to classify surfaces in simply-connect…

Smooth and topological almost concordance

Matthias Nagel, Patrick Orson, JungHwan Park +1

We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo con…

Smoothing 3-manifolds in 5-manifolds

Michelle Daher, Mark Powell

We show that every locally flat topological embedding of a 3-manifold in a smooth 5-manifold is homotopic, by a small homotopy, to a smooth embedding. We deduce that topologically…

Stably diffeomorphic manifolds and the realisation of modified surgery obstructions

Anthony Conway, Diarmuid Crowley, Mark Powell +1

For every $k \geq 2$ we construct infinitely many $4 k$ -dimensional manifolds that are all stably diffeomorphic but pairwise not homotopy equivalent. Each of these manifolds has hyp…

math.GT2013

Covering link calculus and the bipolar filtration of topologically slice links

Jae Choon Cha, Mark Powell

The bipolar filtration introduced by T. Cochran, S. Harvey, and P. Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that…

math.GT2010

An Injectivity Theorem for Casson-Gordon Type Representations relating to the Concordance of Knots and Links

Stefan Friedl, Mark Powell

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let $I ¨ $ be a group and let $M \to N$ be a homomorphi…

The round handle problem

Min Hoon Kim, Mark Powell, Peter Teichner

We present the Round Handle Problem, proposed by Freedman and Krushkal. It asks whether a collection of links, which contains the Generalised Borromean Rings, are slice in a 4-mani…

Cobordisms to weakly splittable links

Stefan Friedl, Mark Powell

We show that if a link L with non-zero Alexander polynomial admits a locally flat cobordism to a `weakly m-split link', then the cobordism must have genus at least (m-1)/2. This ge…

Homotopy ribbon concordance and Alexander polynomials

Stefan Friedl, Mark Powell

We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L then the Alexander polynomial of L divides the Alexander polynomial of J.

Stabilization distance between surfaces

Allison N. Miller, Mark Powell

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary fo…

Some remarks on $h$ -cobordisms between smooth 4-manifolds

Alexander Kupers, Mark Powell

It is not known whether the realisation part of the $s$ -cobordism theorem holds for smooth 4-manifolds, nor whether every pair of smoothly $h$ -cobordant 4-manifolds is also smoothl…

Embedding surfaces in 4-manifolds

Daniel Kasprowski, Mark Powell, Arunima Ray +1

We prove a surface embedding theorem for 4-manifolds with good fundamental group in the presence of dual spheres, with no restriction on the normal bundles. The new obstruction is…

Concordance of links with identical Alexander invariants

Jae Choon Cha, Stefan Friedl, Mark Powell

J. Davis showed that the topological concordance class of a link in the 3-sphere is uniquely determined by its Alexander polynomial for 2-component links with Alexander polynomial…

Concordance invariance of Levine-Tristram signatures of links

Matthias Nagel, Mark Powell

We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants.

Generalised doubles and simple homotopy types of high dimensional manifolds

Csaba Nagy, John Nicholson, Mark Powell

We characterise the set of fundamental groups for which there exist $n$ -manifolds that are $h$ -cobordant (hence homotopy equivalent) but not simple homotopy equivalent, when $n$ is…

Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups

Daniel Kasprowski, Mark Powell, Benjamin Ruppik

We show that for an oriented 4-dimensional PoincarÃ© complex with finite fundamental group, whose 2-Sylow subgroup is abelian with at most 2 generators, the homotopy type is determ…

Infinite homotopy stable class for 4-manifolds with boundary

Anthony Conway, Diarmuid Crowley, Mark Powell

We show that for every odd prime $q$ , there exists an infinite family ${M_{i}}_{i = 1}^{\infty}$ of topological 4-manifolds that are all stably homeomorphic to one another, all the m…

A family of freely slice good boundary links

Jae Choon Cha, Min Hoon Kim, Mark Powell

We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all pre…

Blanchfield forms and Gordian distance

Maciej Borodzik, Stefan Friedl, Mark Powell

Given a link in $S^{3}$ we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinki…

Shrinking of toroidal decomposition spaces

Daniel Kasprowski, Mark Powell

Given a sequence of oriented links L^1,L^2,L^3,... each of which has a distinguished, unknotted component, there is a decomposition of the 3-sphere naturally associated to it, whic…

The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds

Daniel Kasprowski, Mark Powell, Peter Teichner

We consider the role of the Kervaire--Milnor invariant in the classification of closed, connected, spin 4-manifolds, typically denoted by $M$ , up to stabilisation by connected sums…

Four-manifolds up to connected sum with complex projective planes

Daniel Kasprowski, Mark Powell, Peter Teichner

We show that closed, connected 4-manifolds up to connected sum with copies of the complex projective plane are classified in terms of the fundamental group, the orientation charact…

Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials

Stefan Friedl, Takahiro Kitayama, Lukas Lewark +2

We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine-Tristram signatures. Then, as an application of twisted Alexander polynomials, we…

Unknotting nonorientable surfaces

Anthony Conway, Patrick Orson, Mark Powell

Given a nonorientable, locally flatly embedded surface in the $4$ -sphere of nonorientable genus $h$ , Massey showed that the normal Euler number lies in $\lbrace -2h,-2h+4,\ldots,2h…

A second order algebraic knot concordance group

Mark Powell

We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group…

A specious unlinking strategy

Stefan Friedl, Matthias Nagel, Mark Powell

We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and t…

math.GT2011

A Second Order Algebraic Knot Concordance Group

Mark Powell

We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group…

Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance

Allison N. Miller, Mark Powell

We give a formula for the duality structure of the 3-manifold obtained by doing zero-framed surgery along a knot in the 3-sphere, starting from a diagram of the knot. We then use t…

Embedding spheres in knot traces

Peter Feller, Allison N. Miller, Matthias Nagel +3

The trace of $n$ -framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a mani…

The $Z$ -genus of boundary links

Peter Feller, JungHwan Park, Mark Powell

The $Z$ -genus of a link $L$ in $S^{3}$ is the minimal genus of a locally flat, embedded, connected surface in $D^{4}$ whose boundary is $L$ and with the fundamental group of t…

Brunnian exotic surface links in the 4-ball

Kyle Hayden, Alexandra Kjuchukova, Siddhi Krishna +3

This paper investigates the exotic phenomena exhibited by links of disconnected surfaces with boundary that are properly embedded in the 4-ball. Our main results provide two differ…

Algebraic criteria for stable diffeomorphism of spin 4-manifolds

Daniel Kasprowski, Mark Powell, Peter Teichner

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^{2} \times S^{2}$ . For a fixed fundamental group, there are primary, secondary and…

Algorithms in 4-manifold topology

Stefan Bastl, Rhuaidi Burke, Rima Chatterjee +28

We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In p…

Grope metrics on the knot concordance set

Tim D. Cochran, Shelly Harvey, Mark Powell

To a special type of grope embedded in 4-space, that we call an admissible grope, we associate a length function for each real number q at least 1. This gives rise to a family of p…

Smoothing topological pseudo-isotopies of 4-manifolds

Patrick Orson, Mark Powell, Oscar Randal-Williams

Given a closed, smooth 4-manifold $X$ and self-diffeomorphism $f$ that is topologically pseudo-isotopic to the identity, we study the question of whether $f$ is moreover smoothly p…

Non-concordant links with homology cobordant zero framed surgery manifolds

Jae Choon Cha, Mark Powell

We use topological surgery in dimension four to give sufficient conditions for the zero framed surgery manifold of a 3-component link to be homology cobordant to the 3-torus, which…