1,917 research outputs found
Topological classification of affine operators on unitary and Euclidean spaces
We classify affine operators on a unitary or Euclidean space U up to
topological conjugacy. An affine operator is a map f: U-->U of the form
f(x)=Ax+b, in which A: U-->U is a linear operator and b in U. Two affine
operators f and g are said to be topologically conjugate if hg=fh for some
homeomorphism h: U-->U
Surgery untying of coloured knots
For p=3 and for p=5 we prove that there are exactly p equivalence classes of
p-coloured knots modulo (+/-1)--framed surgeries along unknots in the kernel of
a p-colouring. These equivalence classes are represented by connect-sums of n
left-hand (p,2)-torus knots with a given colouring when n=1,2,...,p. This gives
a 3-colour and a 5-colour analogue of the surgery presentation of a knot.Comment: This is the version published by Algebraic & Geometric Topology on 24
May 200
Equivariant characteristic classes of singular complex algebraic varieties
Homology Hirzebruch characteristic classes for singular varieties have been
recently defined by Brasselet-Schuermann-Yokura as an attempt to unify
previously known characteristic class theories for singular spaces (e.g.,
MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and
Goresky-MacPherson L-classes, respectively). In this note we define equivariant
analogues of these classes for singular quasi-projective varieties acted upon
by a finite group of algebraic automorphisms, and show how these can be used to
calculate the homology Hirzebruch classes of global quotient varieties. We also
compute the new classes in the context of monodromy problems, e.g., for
varieties that fiber equivariantly (in the complex topology) over a connected
algebraic manifold. As another application, we discuss Atiyah-Meyer type
formulae for twisted Hirzebruch classes of global orbifolds.Comment: v2: updates include a motivic approach, as well as an Atiyah-Meyer
formula for global orbifolds, including a defect formul
On fibering and splitting of 5-manifolds over the circle
Our main result is a generalization of Cappell's 5-dimensional splitting
theorem. As an application, we analyze, up to internal s-cobordism, the
smoothable splitting and fibering problems for certain 5-manifolds mapping to
the circle. For example, these maps may have homotopy fibers which are in the
class of finite connected sums of certain geometric 4-manifolds. Most of these
homotopy fibers have non-vanishing second mod 2 homology and have fundamental
groups of exponential growth, which are not known to be tractable by
Freedman--Quinn topological surgery. Indeed, our key technique is topological
cobordism, which may not be the trace of surgeries.Comment: 22 pages, exposition revised for better self-containmen
Fredholm-Lagrangian-Grassmannian and the Maslov index
We explain the topology of the space, so called,
Fredholm-Lagrangian-Grassmannain and the quantity ``Maslov index'' for paths in
this space based on the standard theory of Functional Analysis. Our standing
point is to define the Maslov index for arbitrary paths in terms of the
fundamental spectral property of the Fredholm operators, which was first
recognized by J. Phillips and used to define the ``Spectral flow''. We tried to
make the arguments to be all elementary and we summarize basic facts for this
article from Functional Analysis in the Appendix.Comment: 64 pages, no figur
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