19 research outputs found
On the Borsuk conjecture concerning homotopy domination
In the seminal monograph "Theory of retracts", Borsuk raised the following
question: suppose two compact ANR's are --equal, i.e. mutually homotopy
dominate each other, are they homotopy equivalent? The current paper approaches
this question in two ways. On one end, we provide conditions on the fundamental
group which guarantee a positive answer to the Borsuk question. On the other
end, we construct various examples of compact --equal, not homotopy
equivalent continua, with distinct properties. The first class of these
examples has trivial all known algebraic invariants (such as homology, homotopy
groups etc.) The second class is given by --connected continua, for any ,
which are infinite --complexes, and hence ANR's, on a complement of a
point.Comment: 18 pages, 6 figures; final version accepted for publicatio
Surgery and the Spectrum of the Dirac Operator
We show that for generic Riemannian metrics on a simply-connected closed spin
manifold of dimension at least 5 the dimension of the space of harmonic spinors
is no larger than it must be by the index theorem. The same result holds for
periodic fundamental groups of odd order.
The proof is based on a surgery theorem for the Dirac spectrum which says
that if one performs surgery of codimension at least 3 on a closed Riemannian
spin manifold, then the Dirac spectrum changes arbitrarily little provided the
metric on the manifold after surgery is chosen properly.Comment: 23 pages, 4 figures, to appear in J. Reine Angew. Mat
Recommended from our members
Symplectic symmetries of 4-manifolds
A study of symplectic actions of a finite group G on smooth 4-manifolds is initiated. The central new idea is the use of G-equivariant Seiberg–Witten–Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set structure (and the action around it) of a symplectic cyclic action of prime order on a minimal symplectic 4-manifold with . Comparison of this result with the case of locally linear topological actions is made. As an application of these considerations, the triviality of many such actions on a large class of 4-manifolds is established. In particular, we show the triviality of homologically trivial symplectic symmetries of a K3 surface (in analogy with holomorphic automorphisms). Various examples and comments illustrating our considerations are also included
Recommended from our members
Symmetries and exotic smooth structures on a K3 surface
Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and a comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and, as a result, the nonsmoothability of actions induced by a holomorphic automorphism of prime order at least 7 is proved, and the nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 that has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of prime order at least 5 is explicitly determined, provided that the action is homologically nontrivial