arXiv subjects

David Baraglia

Publications and source records attributed to David Baraglia.

At least 19 recordsLinked to original sources

arXiv2026

Exotic diffeomorphisms of reducible $4$-manifolds with odd $b_+$

A diffeomorphism of a $4$-manifold is said to be exotic if it is continuously isotopic to the identity but not smoothly isotopic to the identity. Ruberman constructed the first examples of exotic diffeomorphisms on simply-connected closed $4$-manifolds. His examples were reducible $4$-manifolds that necessarily have even $b_+$ in order that they can be detected by the families Seiberg--Witten or Donaldson invariants. Later Konno and Baraglia produced exotic diffeomorphisms on irreducible $4$-manifolds with odd $b_+$. In this paper, we will construct exotic diffeomorphisms on reducible $4$-manifolds with odd $b_+$. Exoticness is detected using a families Bauer--Furuta invariant. In proving our results we need to work with families moduli spaces which are not framed and so do not give rise to framed cobordism invariants. We overcome this difficulty by considering a Bauer--Furuta type invariant valued in {\em pin-cobordism}. In addition to constructing exotic diffeomorphisms, we also find new examples of simply-connected $4$-manifolds whose mapping class groups are not finitely generated.

math.GT↗

arXiv2026

Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$

We compute the Floer homology and Seiberg-Witten Floer homotopy type of Seifert rational homology $3$-spheres which fiber over $\mathbb{RP}^2$. We show that they are all $L$-spaces and their Floer homotopy type is a suspension of $S^0$. Additionally, we compute the Ozsv\'ath-Szab\'o $d$-invariants, or equivalently the Seiberg-Witten $\delta$-invariants for such $3$-manifolds. This is done by computing the eta invariant of spin$^c$-Dirac operators associated to spin$^c$-connections covering the adiabatic connection, a certain metric connection distinct from the Levi-Civita connection. It turns out that this eta invariant involves a contribution given by the eta invariant of an orbifold pin$^c$-connection on the orbifold base of the Seifert fibration, which we also compute.

math.GT↗

arXiv2025

Exotic diffeomorphisms of 4-manifolds with b_+ = 2

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The mapping class group of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The Torelli group of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \ge 10$, the Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$. We also prove that the mapping class group of $2 \mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg-Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg-Witten invariants plays an important role.

math.GT↗

arXiv2025

A Fourier-Mukai Transform For KR Theory

In complex K-theory, the Fourier-Mukai transform is an isomorphism between K-theory groups of a torus and its dual torus which is defined by pullback, tensoring by the Poincaré line bundle and pushforward. The Fourier-Mukai transform extends to families of dual tori provided one works with twisted K-theory. The Fourier-Mukai transform is then an isomorphism between twisted K-theory groups of $T$-dual torus bundles. In this paper we prove an extension of these results to twisted KR-theory. We introduce a notion of Real $T$-duality for torus bundles with Real structures and prove the existence of Real $T$-duals. We then define a Real Fourier-Mukai transform for Real $T$-dual torus bundles and prove that it is an isomorphism. Lastly, we consider an application of these results to the families index of Real Dirac operators which is relevant to Real Seiberg-Witten theory.

math.KT↗

arXiv2025

An adjunction inequality for Real embedded surfaces

A Real structure on a $4$-manifold $X$ is an orientation preserving smooth involution $\sigma$. We say that an embedded surface $\Sigma \subset X$ is Real if $\sigma$ maps $\Sigma$ to itself orientation reversingly. We prove that a cohomology class $u \in H^2(X ; \mathbb{Z})$ can be represented by a Real embedded surface if and only if $u$ can be lifted to a class in equivariant cohomology $H^2_{\mathbb{Z}_2}(X ; \mathbb{Z}_-)$. We prove that if the Real Seiberg--Witten invariants of $X$ are non-zero then the genus of Real embedded surfaces in $X$ satisfy an adjunction inequality. We prove two versions of the adjunction inequality, one for non-negative self-intersection and one for arbitrary self-intersection. We show with examples that the minimal genus of Real embedded surfaces can be larger than the minimal genus of arbitrary embedded surfaces.

math.GT↗

arXiv2025

Irreducible 4-manifolds can admit exotic diffeomorphisms

We prove that a variety of examples of minimal complex surfaces admit exotic diffeomorphisms, providing the first known instances of exotic diffeomorphisms of irreducible 4-manifolds. We also give sufficient conditions for the boundary Dehn twist on a spin 4-manifold with $S^3$ boundary to be non-trivial in the relative mapping class group. This gives many new examples of non-trivial boundary Dehn twists.

math.GT↗

arXiv2025

Exotic embedded surfaces and involutions from Real Seiberg-Witten theory

Using Real Seiberg--Witten theory, Miyazawa introduced an invariant of certain 4-manifolds with involution and used this invariant to construct infinitely many exotic involutions on $\mathbb{CP}^2$ and infinitely many exotic smooth embeddings of $\mathbb{RP}^2$ in $S^4$. In this paper we extend Miyazawa's construction to a large class of 4-manifolds, giving many infinite families of involutions on 4-manifolds which are conjugate by homeomorphisms but not by diffeomorphisms and many infinite families of exotic embeddings of non-orientable surfaces in 4-manifolds, where exotic means continuously isotopic but not smoothly isotopic. Exoticness of our construction is detected using Real Seiberg--Witten theory. We study Miyazawa's invariant, relate it to the Real Seiberg--Witten invariants of Tian--Wang and prove various fundamental results concerning the Real Seiberg--Witten invariants such as: relation to positive scalar curvature, wall-crossing, a mod 2 formula for spin structures, a localisation formula relating ordinary and Real Seiberg--Witten invariants, a connected sum formula and a fibre sum formula.

math.GT↗

arXiv2024

Constraints on embedded spheres and real projective planes in 4-manifolds from Seiberg-Witten theory

We calculate the Seiberg-Witten invariants of branched covers of prime degree, where the branch locus consists of embedded spheres. Aside from the formula itself, our calculations give rise to some new constraints on configurations of embedded spheres in 4-manifolds. Using similar methods, we also obtain new constraints on embeddings of real projective planes and spheres with a cusp singularity. Moreover, we show that the existence of certain configurations of surfaces would give rise to 4-manifolds of non-simple type. Our proof makes use of equivariant Seiberg-Witten invariants as well as a gluing formula for the relative Seiberg-Witten invariants of 4-manifolds with positive scalar curvature boundary.

math.GT↗

arXiv2024

Equivariant Seiberg-Witten theory

We introduce and study equivariant Seiberg-Witten invariants for $4$-manifolds equipped with a smooth action of a finite group $G$. Our invariants come in two types: cohomological, valued in the group cohomology of $G$ and $K$-theoretic, valued in the representation ring of $G$. We establish basic properties of the invariants such as wall-crossing and vanishing of the invariants for $G$-invariant positive scalar curvature metrics. We establish a relation between the equivariant Seiberg-Witten invariants and families Seiberg-Witten invariants. Sufficient conditions are found under which equivariant transversality can be achieved leading to smooth moduli spaces on which $G$ acts. In the zero-dimensional case this yields a further invariant of the $G$-action valued in a refinement of the Burnside ring of $G$. We prove localisation formulas in cohomology and $K$-theory, relating the equivariant Seiberg-Witten invariants to moduli spaces of $G$-invariant solutions. We give an explicit formula for the invariants for holomorphic group actions on Kähler surfaces. We also prove a gluing formula for the invariants of equivariant connected sums. Various applications and consequences of the theory are considered.

math.DG↗

arXiv2024

Brieskorn spheres, cyclic group actions and the Milnor conjecture

In this paper we further develop the theory of equivariant Seiberg-Witten-Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain the following applications. First, we show that the knot concordance invariants $θ^{(c)}$ defined by the first author satisfy $θ^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever $c$ is a prime not dividing $ab$. Since $θ^{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture of a similar flavour to the proofs using the Ozsváth-Szabó $τ$-invariant or Rasmussen $s$-invariant. Second, we prove that a free cyclic group action on a Brieskorn homology $3$-sphere $Y = Σ(a_1 , \dots , a_r)$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. This generalises to arbitrary $r$ the result of Anvari-Hambleton in the case $r=3$. Third, given a finite subgroup of the Seifert circle action on $Y = Σ(a_1 , \dots , a_r)$ of prime order $p$ acting non-freely on $Y$, we prove that if the rank of $HF_{red}^+(Y)$ is greater than $p$ times the rank of $HF_{red}^+(Y/\mathbb{Z}_p)$, then the $\mathbb{Z}_p$-action on $Y$ does not extend smoothly to any contractible smooth $4$-manifold bounding $Y$. We also prove a similar non-extension result for equivariant connected sums of Brieskorn homology spheres.

math.GT↗

arXiv2024

New invariants of involutions from Seiberg-Witten Floer theory

We study equivariant Seiberg-Witten Floer theory of rational homology $3$-spheres in the special case where the group action is given by an involution. The case of involutions deserves special attention because we can couple the involution to the charge conjugation symmetry of Seiberg-Witten theory. This leads to new Floer-theoretic invariants which we study and apply in a variety of applications. In particular, we construct a series of delta-invariants $δ^E_*, δ^R_*, δ^S_*$ which are the equivariant equivalents of the Ozsváth-Szabó $d$-invariant. The delta-invariants come in three types: equivariant, Real and spin depending on the type of the spin$^c$-structure involved. The delta-invariants satisfy many useful properties, including a Froyshov-type inequality for equivariant cobordisms. We compute the delta-invariants in a wide range of examples including: equivariant plumbings, branched double covers of knots and equivariant Dehn surgery. We also consider various applications including obstructions to extending involutions over bounding $4$-manifolds, non-smoothable involutions on $4$-manifolds with boundary, equivariant embeddings of $3$-manifolds in $4$-manifolds and non-orientable surfaces bounding knots.

math.GT↗

arXiv2024

An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds

Consider a smooth $4$-manifold $X$ and a diffeomorphism $f : X \to X$. We give an obstruction in the form of an adjunction inequality for an embedded surface in $X$ to be isotopic to its image under $f$. It follows that the minimal genus of a surface representing a given homology class and which is isotopic to its image under $f$ is generally larger than the minimal genus without the isotopy condition. We give examples where the inequality is strict. We use our obstruction to construct examples of infinitely many embedded surfaces which are all continuously isotopic but mutually non-isotopic smoothly.

math.DG↗

arXiv2023

On the mapping class groups of simply-connected smooth 4-manifolds

The mapping class group $M(X)$ of a smooth manifold $X$ is the group of smooth isotopy classes of orientation preserving diffeomorphisms of $X$. We prove a number of results about the mapping class groups of compact, simply-connected, smooth $4$-manifolds. We prove that $M(X)$ is non-finitely generated for $X = 2n \mathbb{CP}^2 # 10n \overline{\mathbb{CP}^2}$, where $n \ge 3$ is odd. Let $\Gamma(X)$ denote the group of automorphisms of the intersection lattice of $X$ that can be realised by diffeomorphisms. Then $M(X)$ is an extension of $\Gamma(X)$ by $T(X)$, the Torelli group of isotopy classes of diffeomorphisms that act trivially in cohomology. We prove that this extension is split for connected sums of $\mathbb{CP}^2$, but is not split for $2\mathbb{CP}^2 # n \overline{\mathbb{CP}^2}$, where $n \ge 11$. We prove that the Nielsen realisation problem fails for certain finite subgroups of $M( p \mathbb{CP}^2 # q \overline{\mathbb{CP}^2} )$ whenever $p+q \ge 4$. Lastly we study the extension $M_1(X) \to M(X)$, where $M_1(X)$ is the group of isotopy classes of diffeomorphisms of $X$ which fix a neighbourhood of a point. When $X = K3$ or $K3 # (S^2 \times S^2)$ we prove that $M_1(X) \to M(X)$ is a non-trivial extension of $M(X)$ by $\mathbb{Z}_2$. Moreover, we completely determine the extension class of $M_1(K3) \to M(K3)$.

math.GT↗

arXiv2023

The mod 2 Seiberg-Witten invariants of spin structures and spin families

We completely determine the mod $2$ Seiberg-Witten invariants for any spin structure on any closed, oriented, smooth $4$-manifold $X$. Our computation confirms the validity of the simple type conjecture mod $2$ for spin structures. Our proof also works for families of spin $4$-manifolds and thus computes the mod $2$ Seiberg-Witten invariants for spin families. The proof of our main result uses $Pin(2)$-symmetry to define an enhancement of the mod $2$ Seiberg-Witten invariants. We prove a connected sum formula for the enhanced invariant using localisation in equivariant cohomology. Unlike the usual Seiberg-Witten invariant, the enhanced invariant does not vanish on taking connected sums and by exploiting this property, we are able to compute the enhanced invariant.

math.GT↗

arXiv2023

On the slice genus of quasipositive knots in indefinite 4-manifolds

Let $X$ be a closed indefinite $4$-manifold with $b_+(X) = 3 \; ({\rm mod} \; 4)$ and with non-vanishing mod $2$ Seiberg--Witten invariants. We prove a new lower bound on the genus of a properly embedded surface in $X \setminus B^4$ representing a given homology class and with boundary a quasipositive knot $K \subset S^3$. In the null-homologous case our inequality implies that the minimal genus of such a surface is equal to the slice genus of $K$. If $X$ is symplectic then our lower bound differs from the minimal genus by at most $1$ for any homology class that can be represented by a symplectic surface. Along the way, we also prove an extension of the adjunction inequality for closed $4$-manifolds to classes of negative self-intersection without requiring $X$ to be of simple type.

math.GT↗

arXiv2022

Non-trivial smooth families of $K3$ surfaces

Let $X$ be a complex $K3$ surface, ${\rm Diff}(X)$ the group of diffeomorphisms of $X$ and ${\rm Diff}_0(X)$ the identity component. We prove that the fundamental group of ${\rm Diff}_0(X)$ contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg--Witten invariants. The moduli space of Einstein metrics on $X$ is used as a key ingredient in the proof.

math.DG↗

arXiv2022

Equivariant Seiberg-Witten-Floer cohomology

We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology $3$-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of $d$-invariants $d_{G,c}(Y,\mathfrak{s})$ which are indexed by the group cohomology of $G$. These invariants satisfy a Froyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these $d$-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding $4$-manifolds, Nielsen realisation problems for $4$-manifolds with boundary and obstructions to equivariant embeddings of $3$-manifolds in $4$-manifolds.

math.GT↗

arXiv2022

Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants $θ^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg-Witten-Floer cohomology, constructed by the author and Hekmati, applied to the degree $q$ cyclic cover of $S^3$ branched over $K$. In the case $q=2$, our invariant $θ^{(2)}(K)$ shares many similarities with the knot Floer homology invariant $ν^+(K)$ defined by Hom and Wu. Our invariants $θ^{(q)}(K)$ give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding $K$ in a definite $4$-manifold with boundary $S^3$.

math.GT↗