Page 589 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 589
LOW-DIMENSIONAL TOPOLOGY (MS-11)
Virtual Morse-Bott index, moduli spaces of pairs, and applications to
topology of smooth four-manifolds
Paul Feehan, feehan@math.rutgers.edu
Rutgers University, New Brunswick, United States
Coauthor: Thomas Leness
n 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, al-
most 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 com-
plex, 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.
Non-orientable slice surfaces and inscribed rectangles
Peter Feller, peter.feller@math.ch
ETH Zurich, Switzerland
We consider the complexity of non-orientable locally-flat surfaces in the four-ball B4 and in
S1 × B3 with boundary a prescribed torus knot and discuss differences between the locally-flat
and smooth setup.
Our investigation is motivated by the following old metric problem posed by Toeplitz over
a hundred years ago: Does every Jordan curve (the image of a continuous injection from the
circle to the Euclidean plane), contain four points that form the corners of a square.
Based on joint work with M. Golla.
The slope of a link computed via C-complexes
Ana G. Lecuona, ana.lecuona@glasgow.ac.uk
University of Glasgow, United Kingdom
Together with Alex Degtyarev and Vincent Florence we introduced a new link invariant, called
slope, of a colored link in an integral homology sphere. In this talk I will define the invariant,
highlight some of its most interesting properties as well as its relationship to Conway polyno-
mials and to the Kojima–Yamasaki eta-function. The stress in this talk will be on our latest
computational progress: a formula to calculate the slope from a C-complex.
587
Virtual Morse-Bott index, moduli spaces of pairs, and applications to
topology of smooth four-manifolds
Paul Feehan, feehan@math.rutgers.edu
Rutgers University, New Brunswick, United States
Coauthor: Thomas Leness
n 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, al-
most 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 com-
plex, 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.
Non-orientable slice surfaces and inscribed rectangles
Peter Feller, peter.feller@math.ch
ETH Zurich, Switzerland
We consider the complexity of non-orientable locally-flat surfaces in the four-ball B4 and in
S1 × B3 with boundary a prescribed torus knot and discuss differences between the locally-flat
and smooth setup.
Our investigation is motivated by the following old metric problem posed by Toeplitz over
a hundred years ago: Does every Jordan curve (the image of a continuous injection from the
circle to the Euclidean plane), contain four points that form the corners of a square.
Based on joint work with M. Golla.
The slope of a link computed via C-complexes
Ana G. Lecuona, ana.lecuona@glasgow.ac.uk
University of Glasgow, United Kingdom
Together with Alex Degtyarev and Vincent Florence we introduced a new link invariant, called
slope, of a colored link in an integral homology sphere. In this talk I will define the invariant,
highlight some of its most interesting properties as well as its relationship to Conway polyno-
mials and to the Kojima–Yamasaki eta-function. The stress in this talk will be on our latest
computational progress: a formula to calculate the slope from a C-complex.
587
