26 research outputs found
An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
We prove an analogue of the Kotschick-Morgan conjecture in the context of
SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten
invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The
main technical difficulty in the SO(3)-monopole program relating the
Seiberg-Witten and Donaldson invariants has been to compute intersection
pairings on links of strata of reducible SO(3) monopoles, namely the moduli
spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck
compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032].
In this monograph, we prove --- modulo a gluing theorem which is an extension
of our earlier work in [arXiv:math/9907107] --- that these intersection
pairings can be expressed in terms of topological data and Seiberg-Witten
invariants of the four-manifold. This conclusion is analogous to the
Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson
invariants of a four-manifold with ; that wall-crossing formula and
the resulting structure of Donaldson invariants for four-manifolds with
were established, assuming the Kotschick-Morgan conjecture, by
Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier
[arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the
Kotschick-Morgan conjecture to an extension of previously established gluing
theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and
references therein). Since the first version of our monograph was circulated,
applications of our results have appeared in the proof of Property P for knots
by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson
invariants for symplectic four-manifolds [arXiv:1301.0377].Comment: x + 229 page
Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds
In Feehan and Leness (2020), we introduced an approach to Morse-Bott theory,
called virtual Morse-Bott theory, for Hamiltonian functions of circle actions
on closed, real analytic, almost Hermitian spaces. In the case of Hamiltonian
functions of circle actions on closed, smooth, almost Kaehler (symplectic)
manifolds, virtual Morse-Bott theory coincides with classical Morse-Bott theory
due to Bott (1954) and Frankel (1959). Positivity of virtual Morse-Bott indices
implies downward gradient flow in the top stratum of smooth points in the
analytic space. In this monograph, we apply our method to the moduli space of
SO(3) monopoles over a complex, Kaehler surface, we use the Atiyah-Singer Index
Theorem to compute virtual Morse-Bott indices of all critical strata
(Seiberg-Witten moduli subspaces), and we prove that these indices are positive
in a setting motivated by the conjecture that all closed, smooth four-manifolds
of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.Comment: 167+xii pages, 1 figure. Supporting background material drawn from
our monograph arXiv:math/020304
PU(2) monopoles and a conjecture of Marino, Moore, and Peradze
In this article we show that some of the recent results of Marino, Moore, and
Peradze (math.DG/9812042, hep-th/9812055) -- in particular their conjecture
that all closed, smooth four-manifolds with b_2^+ > 1 (and Seiberg-Witten
simple type) are of `superconformal simple type' -- can be understood using a
simple mathematical argument via the PU(2)-monopole cobordism of Pidstrigach
and Tyurin (dg-ga/9507004) and results of the first and third authors
(dg-ga/9712005, dg-ga/9709022).Comment: 13 pages, 1 figure. Improved exposition, typographical slips
corrected, figure and references added. Minor correction on page 2. To appear
in Mathematical Research Letter
SO(3) Monopoles, Level-One Seiberg-Witten Moduli Spaces, and Witten's Conjecture in Low Degrees
We prove Witten's formula relating the Donaldson and Seiberg-Witten series
modulo powers of degree , with , for
four-manifolds obeying some mild conditions, where and are
their Euler characteristic and signature. We use the moduli space of SO(3)
monopoles as a cobordism between a link of the Donaldson moduli space of
anti-self-dual SO(3) connections and links of the moduli spaces of
Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions
from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of
the Uhlenbeck compactification of the moduli space of SO(3) monopoles.Comment: Minor corrections; details added. Topology and its Applications, 91
pages, to appea