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Mark Powell

Publications and source records attributed to Mark Powell.

At least 37 records · Page 2Linked to original sources
arXiv2024

Unknotting nonorientable surfaces

Given a nonorientable, locally flatly embedded surface in the $4$-sphere of nonorientable genus $h$, Massey showed that the normal Euler number lies in $\lbrace -2h,-2h+4,\ldots,2h-4,2h \rbrace$. We prove that every such surface with knot group of order two is topologically unknotted, provided that the normal Euler number is not one of the extremal values in Massey's range. When $h$ is $1$, $2$, or $3$, we prove the same holds even with extremal normal Euler number. We also study nonorientable embedded surfaces in the 4-ball with boundary a knot $K$ in the 3-sphere, again where the surface complement has fundamental group of order two and nonorientable genus $h$. We prove that any two such surfaces with the same normal Euler number become topologically isotopic, rel. boundary, after adding a single tube to each. If the determinant of $K$ is trivial, we show that any two such surfaces are isotopic, rel. boundary, again provided that they have non-extremal normal Euler number, or that $h$ is $1$, $2$, or $3$.

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arXiv2024

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

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 with copies of $S^2 \times S^2$. This stable classification is detected by a spin bordism group over the classifying space $Bπ$ of the fundamental group. Part of the computation of this bordism group via an Atiyah--Hirzebruch spectral sequence is determined by a collection of codimension two Arf invariants. We show that these Arf invariants can be computed by the Kervaire--Milnor invariant evaluated on certain elements of $π_2(M)$. In particular this yields a new stable classification of spin $4$-manifolds with 2-dimensional fundamental groups, namely those for which $Bπ$ admits a finite 2-dimensional CW-complex model.

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arXiv2024

Generalised doubles and simple homotopy types of high dimensional manifolds

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 sufficiently large. In particular, for $n \ge 12$ even, we show that examples exist for any finitely presented group $G$ such that the involution on the Whitehead group $Wh(G)$ is nontrivial. This expands on previous work, where we constructed the first examples of even-dimensional manifolds that are homotopy equivalent but not simple homotopy equivalent. Our construction is based on doubles of thickenings, and a key ingredient of the proof is a formula for the Whitehead torsion of a homotopy equivalence between such manifolds.

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arXiv2024

$4$-manifolds with boundary and fundamental group $\mathbb{Z}$

We classify topological $4$-manifolds with boundary and fundamental group $\mathbb{Z}$, under some assumptions on the boundary. We apply this to classify surfaces in simply-connected $4$-manifolds with $S^3$ boundary, where the fundamental group of the surface complement is $\mathbb{Z}$. We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every Hermitian form over $\mathbb{Z}[t^{\pm 1}]$ arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group $\mathbb{Z}$. For surfaces we have a similar result, and in particular we show that every $2$-handlebody with $S^3$ boundary contains a pair of exotic discs.

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arXiv2024

Mapping class groups of simply connected 4-manifolds with boundary

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 smooth, we deduce an analogous description of the stable smooth mapping class group, extending work of Saeki.

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arXiv2024

Stably diffeomorphic manifolds and the realisation of modified surgery obstructions

For every $k \geq 2$ we construct infinitely many $4k$-dimensional manifolds that are all stably diffeomorphic but pairwise not homotopy equivalent. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In fact we construct infinitely many such infinite sets. To achieve this we prove a realisation result for appropriate subsets of Kreck's modified surgery monoid $\ell_{2q+1}(\mathbb{Z}[π])$, analogous to Wall's realisation of the odd-dimensional surgery obstruction $L$-group $L_{2q+1}^s(\mathbb{Z}[π])$.

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arXiv2024

Algebraic criteria for stable diffeomorphism of spin 4-manifolds

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 tertiary obstructions, which together with the signature lead to a complete stable classification. The primary obstruction exactly detects $\mathbb{CP}^2$-stable diffeomorphism and was previously related to algebraic invariants by Kreck and the authors. In this article we formulate conjectural relationships of the secondary and tertiary obstructions with algebraic invariants: the secondary obstruction should be determined by the (stable) equivariant intersection form and the tertiary obstruction via a $τ$-invariant recording intersection data between 2-spheres, with trivial algebraic self-intersection, and their Whitney discs. We prove our conjectures for the following classes of fundamental groups: groups of cohomological dimension at most 3, right-angled Artin groups, abelian groups, and finite groups with quaternion or abelian 2-Sylow subgroups. We apply our theory to give a complete algebraic stable classification of spin $4$-manifolds with fundamental group $\mathbb{Z} \times \mathbb{Z}/2$.

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arXiv2024

Examples of topologically unknotted tori

We show that certain smooth tori with group $\mathbb{Z}$ in $S^4$ have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all $\mathbb{Z}$-surfaces in $S^4$ are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of $S^4$ branched along any of these surfaces is a potentially exotic copy of $S^2 \times S^2$, and, in the case of turned twisted tori, we show they cannot be distinguished from $S^2 \times S^2$ using Seiberg-Witten invariants.

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arXiv2024

Simple spines of homotopy 2-spheres are unique

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 the $2$-sphere has abelian fundamental group. We prove that if two simple spines represent the same generator of $H_2(X)$ then they are ambiently isotopic. In particular, the theorem applies to simple shake-slicing $2$-spheres in knot traces.

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arXiv2024

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

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 references are given, with indications of the strategies employed in the proofs. Where appropriate we give statements for manifolds of all dimensions. Many intuitively plausible theorems which are standard results in differential topology are either extraordinarily deep results in the topological category, are open, or are known to be false. Hence one must proceed with caution. This book seeks to help 4-manifold topologists navigate potential pitfalls, and to apply the many powerful results that do exist with confidence.

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arXiv2023

Brunnian exotic surface links in the 4-ball

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 different constructions of exotic pairs of surface links that are Brunnian, meaning that all proper sublinks of the surface are trivial. We then modify these core constructions to vary the number of components in the exotic links, the genera of the components, and the number of components that must be removed before the surfaces become unlinked. Our arguments extend two tools from 3-dimensional knot theory into the 4-dimensional setting: satellite operations, especially Bing doubling, and covering links in branched covers.

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arXiv2023

Infinite homotopy stable class for 4-manifolds with boundary

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 manifolds $M_i$ have isometric rank one equivariant intersection pairings and boundary $L(2q, 1) # (S^1 \times S^2)$, but they are pairwise not homotopy equivalent via any homotopy equivalence that restricts to a homotopy equivalence of the boundary.

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arXiv2023

Characterisation of homotopy ribbon discs

Let $Γ$ be either the infinite cyclic group $\mathbb{Z}$ or the Baumslag-Solitar group $\mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$. Let $K$ be a slice knot admitting a slice disc $D$ in the 4-ball whose exterior has fundamental group $Γ$. We classify the $Γ$-homotopy ribbon slice discs for $K$ up to topological ambient isotopy rel. boundary. In the infinite cyclic case, there is a unique equivalence class of such slice discs. When $Γ$ is the Baumslag-Solitar group, there are at most two equivalence classes of $Γ$-homotopy ribbon discs, and at most one such slice disc for each lagrangian of the Blanchfield pairing of $K$.

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arXiv2023

Counterexamples in 4-manifold topology

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 and how they are related to one another. We explain implications e.g. that $h$-cobordant manifolds are stably homeomorphic, and we provide examples illustrating the failure of other potential implications. The information is conveniently organised in a flowchart and a table.

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arXiv2023

Embedding surfaces in 4-manifolds

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 a Kervaire-Milnor invariant for surfaces and we give a combinatorial formula for its computation. For this we introduce the notion of band characteristic surfaces.

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arXiv2023

Gluck twists on concordant or homotopic spheres

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 the Gluck twist operation on $M$ along $S$ and $T$ respectively. We show that if $S$ and $T$ are concordant, then $M_S$ and $M_T$ are $s$-cobordant, and so if $π_1(M)$ is good, then $M_S$ and $M_T$ are homeomorphic. Similarly, if $S$ and $T$ are homotopic then we show that $M_S$ and $M_T$ are simple homotopy equivalent. Under some further assumptions, we deduce that $M_S$ and $M_T$ are homeomorphic. We show that additional assumptions are necessary by giving an example where $S$ and $T$ are homotopic but $M_S$ and $M_T$ are not homeomorphic. We also give an example where $S$ and $T$ are homotopic and $M_S$ and $M_T$ are homeomorphic but not diffeomorphic.

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arXiv2022

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

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 invertible slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum $\#^n K$ is at least $n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.

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