arXiv subjects
Publications and source records attributed to Juan Muñoz-Echániz.
We study a class of diffeomorphisms of closed manifolds obtained from Davis' reflection trick construction, which are obtained by reflecting a diffeomorphism supported in the fundamental chamber of such a manifold. We use this construction to exhibit the first examples of closed aspherical $4$-manifolds with diffeomorphisms that are topologically isotopic to the identity map, but not smoothly isotopic to it.
We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplectomorphisms which are smoothly non-trivial. In particular, their symplectic Torelli mapping class group is not generated by squared Dehn-Seidel twists on Lagrangian spheres -- providing a negative answer to a question of Donaldson. (2) We provide the first examples of irreducible closed $4$-manifolds (both symplectic and non-symplectic) that admit exotic diffeomorphisms given by Seifert-fibered Dehn twist.
We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of the function, built from a normal crossing compactification of the variety. They are canonically isomorphic to the homology of vanishing cycles and -- in the absence of bifurcations at infinity -- recover the hypercohomology of the perverse sheaf of vanishing cycles, studied extensively in singularity theory and enumerative geometry. Our construction arises as a special case of a more general construction of Morse homology of non-compact manifolds that admit a compactification by a manifold with corners.
We study Dehn--Seidel twists on configurations of Lagrangian spheres in symplectic $K3$ surfaces, using tools from Seiberg--Witten theory. In the case of $ADE$ configurations of Lagrangian spheres, we prove that a naturally associated representation of the generalised Braid group in the symplectic mapping class group is always faithful after abelianising, in a suitable sense. More generally, we prove that squared Dehn--Seidel twists on homologically-distinct Lagrangian spheres are algebraically independent in the abelianisation of the smoothly-trivial symplectic mapping class group, and deduce from this new infinite-generation results. Beyond symplectic $K3$ surfaces, we also establish analogues of these results at the level of the fundamental group of the space of symplectic forms.
We study the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology $3$-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the $SL(2,\mathbb{C})$ Chern--Simons functional and prove a localisation result: the perturbed critical points either approach a compact subset of the $SL(2, \mathbb{C})$ character variety or else `escape to infinity'. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localising critical points. As an application, we obtain formulae for the Euler characteristic and Poincaré polynomial of the stable locus of the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology $3$-sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by four) of any weighted-homogeneous isolated complete intersection singularity whose link is the given $3$-manifold.
We exhibit the first examples of exotic contactomorphisms with infinite order as elements of the contact mapping class group. These are given by certain Dehn twists on the separating sphere in a connected sum of two closed contact 3-manifolds. We detect these by a combination of hard and soft techniques. On the one hand, we make essential use of an invariant for families of contact structures which generalises the Kronheimer--Mrowka contact invariant in monopole Floer homology. We then exploit an h-principle for families of convex spheres in tight contact 3-manifolds, from which we establish a parametric version of Colin's decomposition theorem. As a further application, we also exhibit new exotic 1-parametric phenomena in overtwisted contact 3-manifolds.
We prove that the monodromy diffeomorphism of a complex 2-dimensional isolated hypersurface singularity of weighted-homogeneous type has infinite order in the smooth mapping class group of the Milnor fiber, provided the singularity is not a rational double point. This is a consequence of our main result: the boundary Dehn twist diffeomorphism of an indefinite symplectic filling of the canonical contact structure on a negatively-oriented Seifert-fibered rational homology 3-sphere has infinite order in the smooth mapping class group. Our techniques make essential use of analogues of the contact invariant in the setting of $\mathbb{Z}/p$-equivariant Seiberg--Witten--Floer homology of 3-manifolds.
We use monopole Floer homology to study the topology of the space of contact structures on a 3-manifold. Our main tool is a generalisation of the Kronheimer--Mrowka--Ozsváth--Szabó contact invariant to an invariant for families of contact structures, and we establish foundational results that describe the interaction between this invariant and the module structure in monopole Floer homology. We apply these results in several examples of contact manifolds, such as the links of non-rational surface singularities, to deduce several applications. Namely, we are able to obstruct the existence of sections of a natural fibration over the 2-sphere whose total space is the space of contact structures on the 3-manifold, and from this we are able to detect the existence of exotic loops of contact structures on contact 3-manifolds with convex sphere boundary.
We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More precisely, we establish infinite order non-triviality results for boundary Dehn twists on indefinite symplectic fillings of links of minimally elliptic surface singularities. Using this, we exhibit a wide variety of new phenomena in dimension four: (1) smoothings of isolated complex surface singularities whose Milnor fibration has monodromy with infinite order as a diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3) compactly supported exotic diffeomorphisms of exotic $\mathbb{R}^4$'s and contractible manifolds.