Abstract. There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that smooth free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which the actions extend locally linearly but not smoothly.
Let X 0 denote a compact, simply-connected smooth 4-manifold with boundary the Poincaré homology 3-sphere Σ(2, 3, 5) and with even negative definite intersection form Q X0 = E 8 . We show that free Z/p actions on Σ(2, 3, 5) do not extend to smooth actions on X 0 with isolated fixed points for any prime p > 7. The approach is to study the equivariant version of the Yang-Mills instanton-one moduli space for 4-manifolds with cylindrical ends. As an application we show that for p > 7 a smooth Z/p action on # 8 S 2 × S 2 with isolated fixed points does not split along a free action on Σ(2, 3, 5). The results hold for p = 7 if the action is homologically trivial.
The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres.As an application, we compute the generators and Floer gradings in the singular instanton chain complex of (p, q)-torus knots with odd p and q.where θ : π 1 (Y ) → U(n) is the trivial representation. It is independent of the metric and defines a diffeomorphism invariant of (Y, α).This paper deals with equivariant analogues of the η-and ρ-invariants in presence of an orientation preserving isometric involution τ : Y → Y as defined in [APS75] and [Don78]. Suppose that a representation α : π 1 (Y ) → U(n) is such that its pull-back τ * α is conjugate to α. Then the pull-back of the flat vector bundle E α is isomorphic to E α hence τ can be lifted to a flat bundle automorphism ν : E α → E α . Note that the lift ν need not be an involution and that there may be more than one lift. The lift ν acts on Ω ev (Y ; E α ) via pull-back of forms making the diagramcommute. Each eigenspace W λ,α of the operator B ev α is then ν * -invariant, and the function
Given a 4-manifold with a homologically trivial and locally-linear cyclic group action, we obtain necessary and sufficient conditions for the existence of equivariant bundles. The conditions are derived from the twisted signature formula and are in the form of congruence relations between the fixed point data and the isotropy representations.This is a well-known question of Allan Edmonds. Note that results of Edmonds and Ewing [12] imply that topological locally linear examples exist for odd primes.2. Does there exist a smooth Z/p-action on a homotopy K3 surface, which contains an invariant embedded Brieskorn homology 3-sphere Σ(2, 3, 7) ?
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