We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R² has non-negative isotropic curvature.

Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W^{1, p}(R^p+1, N) be the Sobolev space of measurable maps from R^p+1 into N whose gradients are in L^p. The restriction of u to almost every p-dimensional sphere S in R^p+1 is in W^{1, p}(S, N) and defines an homotopy class in π_p(N) (White 1988). Evaluating a fixed element z of Hom(π_p(N), R) on this homotopy class thus gives a real number Φ_{z, u}(S). The main result of the paper is that any W^{1, p}-weakly convergent limit u of a sequence of smooth maps in C^∞(R^p+1, N), Φ_{z, u} has a rectifiable Poincaré dual$ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $. Here Γ is a a countable union of C¹ curves in R^p+1 with Hausdorff $ {\user1{\mathcal{H}}}^{1} $-measurable orientation $ {\overrightarrow{\Gamma }} :\Gamma \to S^{p} $ and density function θ: Γ→R. The intersection number between $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ and S evaluates Φ_{z, u}(S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n_z, depending only on homotopy operation z, such that $ {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } $ even though the mass $ {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } $ may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n_z) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to R^m for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.

We consider amalgamated free product II₁ factors M = M_1*BM_2*B… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitarily conjugated into one of the M_i’s. We apply this to the case where the M_i’s are w-rigid II₁ factors, with B equal to either C, to a Cartan subalgebra A in M_i, or to a regular hyperfinite II₁ subfactor R in M_i, to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N_1 * CN_2*C…)^t, for some t > 0 and some other similar inclusions of algebras C ⊂ N_i then, after a permutation of indices, (B ⊂ M_i) is inner conjugate to (C ⊂ N_i)^t, for all i. Taking B = C and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $, with {t_i}_i⩾1 = S a given countable subgroup of R₊^*, we obtain continuously many non-stably isomorphic factors M with fundamental group $ {\user1{\mathcal{F}}}{\left( M \right)} $ equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ and Out(M) abelian and calculable. Taking B = R, we get examples of factors with $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $, Out(M) = K, for any given separable compact abelian group K.