The fibre of the degree 3 map, Anick spaces and the double suspension

Let S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

The General Theory of Relativity and Gravitation

The intrinsic curvature of a metric space is captured by the Riemann curvature tensor, which can be contracted to the Ricci tensor and the Ricci scalar. Einstein took these curvature quantities and constructed the Einstein field equations that relate the curvature of space-time to energy and mass density. For an isotropic density, a solution to the field equations is the Schwarzschild metric, which contains mass terms that modify both the temporal and the spatial components of the invariant element. Consequences of the Schwarzschild metric include gravitational time dilation, length contraction, and redshifts. Trajectories in curved space-time are expressed as geodesics through the Schwarzschild metric space. Solutions to the geodesic equation lead to the precession of the perihelion of Mercury and to the deflection of light by the Sun.

Quasi-Poisson Manifolds

A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

Projective characters of degree one and the inflation-restriction sequence

Let G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

On Fixed Points and Multiparameter Ergodic Theorems in Banach Lattices

We present here multiparameter results about positive operators acting on a weakly sequentially complete Banach lattice. Sections 1, 2 and 3 generalize results obtained by M. A. Akcoglu and the second author in the case of a contraction. Even in that case, the classical L1 theory extends to Banach lattices only under an additional monotonicity assumption (C), introduced in [3], without which the TL (or stochastic) ergodic theorem fails. The example proving this in [4] also shows that, without (C), the decomposition of the space into the “positive” part P, the largest support of a T-invariant element, and the “null” part N on which the TL limit is zero (see, e.g., [22], p. 141), also fails. If T is not a contraction but only mean-bounded, then the space decomposes into the “remaining” part Y, the largest support of a T*-invariant element, and the “disappearing part“ Z (see, e.g., [22], p. 172). Here we obtain, for Banach lattices and in the multiparameter case, a unified proof of both decompositions, and of the TL ergodic theorem.

Amenability and invariant subspaces

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

Bounded and invariant elements in 2-firs

1. Introduction. In a principal ideal domain R, any two-sided ideal is of the form Rc = cR, i.e. it has an invariant element as generator, and the customary development of ideal theory in a principal ideal domain (cf. e.g. (10), ch. III) takes on a more transparent form when expressed in terms of invariant elements. Likewise, the one-sided bounded ideals may be studied in terms of their generators.