Searcharxiv⌕ Search

arXiv subjects

Daniel Kasprowski

Publications and source records attributed to Daniel Kasprowski.

At least 19 recordsLinked to original sources
arXiv2026

Homeomorphisms of surfaces in $4$-manifolds

This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary. We prove that most projective planes with knot group $\mathbb{Z}_2$ and the same Euler number are determined by the equivariant intersection form of their exterior. We also prove that knots with prime power determinants bound at most one Moebius band in $D^4$ with knot group $\mathbb{Z}_2$ and a given Euler number. Cancellation results lead to new criteria for homologous discs to be equivalent rel. boundary. Finally, we determine the topological extendable mapping class group of knotted surfaces with abelian knot group.

math.GT↗
arXiv2026

Stably exotic fillings of 3-manifolds

We investigate which 3-manifolds bound 4-manifolds that are homeomorphic but not stably diffeomorphic, where stabilising means taking connected sum with copies of $S^2\times S^2$. We show that every closed, orientable 3-manifold admits such fillings, as do certain families of nonorientable 3-manifolds. In contrast we show that for a 3-manifold containing a 2-sided $\mathbb{RP}^2$, any two smooth, homeomorphic fillings are stably diffeomorphic.

math.GT↗
arXiv2025

The first relative k-invariant

Motivated by work on the homotopy classification of $4$-manifolds with boundary, we define a relative $k$-invariant for pairs of spaces that are homotopy equivalent to CW pairs. We show that for such a pair $(X,Y)$ with Postnikov $2$-type $X \to P_2(X)$, the relative $k$-invariant is the obstruction to the existence of a section $B\pi_1(X)\to P_2(X)$ extending $Y \hookrightarrow X \to P_2(X)$. Given CW pairs $(X_0,Y_0)$ and $(X_1,Y_1)$, as well as a map $h \colon Y_0 \to Y_1$, we also prove that relative $k$-invariants provide a complete obstruction to constructing a map $X_0^{(3)} \cup Y_0 \to X_1$ that extends $h$ and induces given isomorphisms on $\pi_1$ and $\pi_2$.

math.GT↗
arXiv2025

4-manifolds with a given boundary

This paper studies the homotopy and homeomorphism classifications of $4$-manifolds with boundary. Given $4$-manifolds $X_0$ and $X_1$ with fundamental group $\pi$, we consider the problem of extending a homotopy equivalence $h \colon \partial X_0 \to \partial X_1$ to a homotopy equivalence $X_0 \to X_1$. We solve this problem in broad settings for a class of groups that includes free groups, finite cyclic groups, finite dihedral groups, solvable Baumslag-Solitar groups, and many $3$-manifold groups. When the fundamental group is additionally assumed to be good, we use surgery theory to list situations when a homeomorphism $h\colon\partial X_0 \to\partial X_1$ extends to a homeomorphism $X_0 \to X_1$. The outcome recovers results of Boyer in the simply-connected case and work of the first author and Powell when $\pi \cong \mathbb{Z}$ and the $\partial X_i$ have torsion Alexander module.

math.GT↗
arXiv2025

Stably exotic 4-manifolds

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 copies of $S^2 \times S^2$. Orientable stable exotica do not exist by a result of Gompf, but Kreck showed that nonorientable examples are plentiful. We investigate which values of the fundamental group $\pi$ and the first and second Stiefel-Whitney classes $w_1$ and $w_2$ admit stably exotic pairs, providing a complete description if $H_5(\pi;\mathbb{Z})=0$. In particular we produce new stable exotica, and new settings in which they do not arise.

math.GT↗
arXiv2025

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

We give a criterion on a group $\pi$ and a homomorphism $w \colon \pi \to C_2$ under which closed $4$-manifolds with fundamental group $\pi$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic $2$-types. We verify the criterion for a large class of $3$-manifold groups and orientation characters, in particular for the fundamental group $\pi$ of any closed, orientable $3$-manifold whose finite subgroups are cyclic, provided $w$ vanishes on every element of $\pi$ of finite order. We deduce a homeomorphism classification of closed, orientable $4$-manifolds with infinite dihedral fundamental group $\mathbb{Z}/2 * \mathbb{Z}/2$.

math.GT↗
arXiv2024

Stable equivalence relations on 4-manifolds

Kreck's modified surgery gives an approach to classifying smooth $2n$-manifolds up to stable diffeomorphism, i.e. up to connected sum with copies of $S^n \times S^n$. In dimension 4, we use a combination of modified and classical surgery to study various stable equivalence relations which we compare to stable diffeomorphism. Most importantly, we consider homotopy equivalence up to stabilisation with copies of $S^2 \times S^2$. As an application, we show that closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic. We give analogues of the cancellation theorems of Hambleton--Kreck for stable homeomorphism for homotopy up to stabilisations. Finally, we give a complete algebraic obstruction to the existence of closed smooth 4-manifolds which are homotopy equivalent but not simple homotopy equivalent up to connected sum with $S^2 \times S^2$.

math.GT↗
arXiv2022

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 $\pi_1(M)$ 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.

math.GT↗
arXiv2022

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.

math.GT↗
arXiv2022

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.

math.GT↗
arXiv2021

On the Farrell-Jones conjecture for localising invariants

We show the Farrell-Jones conjecture with coefficients in left-exact $\infty$-categories for finitely $\mathcal{F}$-amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over $\mathbb{E}_{1}$-ring spectra.

math.KT↗
arXiv2021

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 $B\pi$ 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 $\pi_2(M)$. In particular this yields a new stable classification of spin $4$-manifolds with 2-dimensional fundamental groups, namely those for which $B\pi$ admits a finite 2-dimensional CW-complex model.

math.GT↗
arXiv2020

Homotopy classification of 4-manifolds whose fundamental group is dihedral

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3). An important class of examples are elliptic surfaces with finite fundamental group.

math.GT↗
arXiv2020

Topological 4-manifolds with 4-dimensional fundamental group

Let $\pi$ be a group satisfying the Farrell-Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincar\'e duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1 and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby-Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby--Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel's conjecture, and simply connected manifolds where rigidity is a consequence of Freedman's classification results.

math.GT↗
arXiv2020

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 $\tau$-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 $\mathbb{Z} \times \mathbb{Z}/2$.

math.GT↗
arXiv2019

Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

We associate to every $G$-bornological coarse space $X$ and every left-exact $\infty$-category with $G$-action a left-exact infinity-category of equivariant $X$-controlled objects. Postcomposing with algebraic K-theory leads to {new} equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact $\infty$-categories.

math.KT↗