3,594 research outputs found
On higher rank instantons & the monopole cobordism program
Witten's conjecture suggests that the polynomial invariants of Donaldson are
expressible in terms of the Seiberg-Witten invariants if the underlying
four-manifold is of simple type. A higher rank version of the Donaldson
invariants was introduced by Kronheimer. Before even having been defined, the
physicists Mari\~no and Moore had already suggested that there should be a
generalisation of Witten's conjecture to this type of invariants. We study a
generalisation of the classical cobordism program to the higher rank situation
and obtain vanishing results which gives evidence that the generalisation of
Witten's conjecture should hold.Comment: This manuscript fusions the two previous manuscripts "What to expect
from U(n) monopoles" and "PU(N) monopoles, higher rank instantons, and the
monopole invariants". Furthermore, one of the vanishing arguments is made
more precise. 29 page
A class of knots with simple representations
We call a knot in the 3-sphere -simple if all representations of the
fundamental group of its complement which map a meridian to a trace-free
element in are binary dihedral. This is a generalisation of being a
2-bridge knot. Pretzel knots with bridge number are not
-simple. We provide an infinite family of knots with bridge number
which are -simple.
One expects the instanton knot Floer homology of a
-simple knot to be as small as it can be -- of rank equal to the knot
determinant . In fact, the complex underlying is of
rank equal to , provided a genericity assumption holds that is
reasonable to expect. Thus formally there is a resemblance to strong L-spaces
in Heegaard Floer homology. For the class of -simple knots that we
introduce this formal resemblance is reflected topologically: The branched
double covers of these knots are strong L-spaces. In fact, somewhat
surprisingly, these knots are alternating. However, the Conway spheres are
hidden in any alternating diagram.
With the methods we use, we show that an integer homology 3-sphere which is a
graph manifold always admits irreducible representations of its fundamental
group.Comment: 22 pages, 10 figures, to appear in Selecta Mathematic
A note on Logarithmic Transformations on the Hopf surface
In this note we study logarithmic transformations in the sense of
differential topology on two fibers of the Hopf surface. It is known that such
transformations are susceptible to yield exotic smooth structures on
four-manifolds. We will show here that this is not the case for the Hopf
surface, all integer homology Hopf surfaces we obtain are diffeomorphic to the
standard Hopf surface.Comment: 9 page
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