1,124 research outputs found
Unfolded Seiberg-Witten Floer spectra, II: Relative invariants and the gluing theorem
We use the construction of unfolded Seiberg-Witten Floer spectra of general
3-manifolds defined in our previous paper to extend the notion of relative
Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main
purposes of this paper is to give a detailed proof of the gluing theorem for
the relative invariants.Comment: 75 pages. Comments are welcomed. v3. Typos fixed. To appear in
Journal of Differential Geometr
On the monopole Lefschetz number of finite order diffeomorphisms
Let be a knot in an integral homology 3-sphere , and the
corresponding -fold cyclic branched cover. Assuming that is a
rational homology sphere (which is always the case when is a prime power),
we give a formula for the Lefschetz number of the action that the covering
translation induces on the reduced monopole homology of . The proof
relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds
and of various -invariants. We give several applications of our formula:
(1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the
mapping tori of all semi-free actions of on integral homology 3-spheres;
(2) we give a novel obstruction (in terms of the Jones polynomial) for the
branched cover of a knot in being an -space; (3) we give a new set of
knot concordance invariants in terms of the monopole Lefschetz numbers of
covering translations on the branched covers.Comment: 39 page, 2 figures. Added a reference to Langte Ma's paper
arXiv:1909.01533, which contains an independent proof of our Theorem B. Final
version, to appear in Geometry and Topolog
Isotopy of the Dehn twist on K3#K3 after a single stabilization
Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the
neck of is not smoothly isotopic to the identity. This provides a new
example of self-diffeomorphisms on 4-manifolds that are isotopic to the
identity in the topological category but not smoothly so. (The first such
examples were given by Ruberman.) In this paper, we use the Pin(2)-equivariant
Bauer-Furuta invariant to show that this Dehn twist is not smoothly isotopic to
the identity even after a single stabilization (connected summing with the
identity map on ). This gives the first example of exotic
phenomena on simply connected smooth 4-manifolds that do not disappear after a
single stabilization.Comment: 19 pages. Version 2:Added the reference to Gompf and Kreck's theorem
and the reference to Szymik's work. Version 3: corrected several typos.
Version 4: Added a few references. Version 5: corrected a few typo
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