In this paper, we define and study an equivariant analogue of Cohen, Farber and Weinberger’s parametrized topological complexity. We show that several results in the non-equivariant case can be extended to the equivariant case. For example, we establish the fibrewise equivariant homotopy invariance of the sequential equivariant parametrized topological complexity. We obtain several bounds on sequential equivariant topological complexity involving the equivariant category. We also obtain the cohomological lower bound and the dimension-connectivity upper bound on the sequential equivariant parametrized topological complexity. In the end, we use these results to compute the sequential equivariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations and some equivariant fibrations involving generalized projective product spaces.

We study moduli spaces of d-dimensional manifolds with embedded particles and discs, which we refer to as decorations. These spaces admit a model in which points are unparametrised d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs constrained to it. We compare this to the space of d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs that are no longer constrained, i.e. the decorations are decoupled. We show that under certain conditions these spaces cannot be distinguished by homology groups within a range. This generalises work by Bödigheimer–Tillmann for oriented surfaces to different tangential structures and also to higher dimensional manifolds. We also extend this result to moduli spaces with more general submanifolds as decorations and specialise in the case of decorations being embedded circles.

Given any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.

A surface $\sum$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration, $G\left( \pi \right)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\text{ }\!\!\pi\!\!\text{ }$ is not an area form on the 2-sphere, then $G\left( \pi \right)$ is diffeomorphic to the cotangent bundle $T*\sum$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

For $p\ge 2$ we introduce the notion of an almost $p$-structure on vector-bundles which generalizes the notion of an almost-complex structure and investigate the existence of almost $p$-structures on spheres and complex projective spaces.

In this paper we show that the Lie-group $\text{Pi}{{\text{n}}_{4}}$ is isomorphic to the semidirect product $\text{(S}{{\text{U}}_{2}}\times \text{S}{{\text{U}}_{2}})\text{Z/2}$ where $Z/2$ operates by flipping the factors. Using this structure theorem we prove a classification theorem for $\text{Pi}{{\text{n}}_{4}}$-bundles over a finite 4-complex $X$.

Let G be a locally compact second countable group, let X be a locally compact second countable Hausdorff space, and view C(X, T) as a trivial G-module. For G countable discrete abelian, we construct an isomorphism between the Moore cohomology group Hn(G, C(X, T)) and the direct sum Ext(Hn-1(G), Ȟl(βX, Ζ)) ⊕ C(X, Hn(G, T)); here Ȟ1 (βX, Ζ) denotes the first Čech cohomology group of the Stone-Čech compactification of X, βX, with integer coefficients. For more general locally compact second countable groups G, we discuss the relationship between the Moore group H2(G, C(X, T)), the set of exterior equivalence classes of element-wise inner actions of G on the stable continuous trace C*-algebra C0(X) ⊗ 𝒦, and the equivariant Brauer group BrG(X) of Crocker, Kumjian, Raeburn, and Williams. For countable discrete abelian G acting trivially on X, we construct an isomorphism is the group of equivalence classes of principal Ĝ bundles over X first considered by Raeburn and Williams.

compact Riemannian 4-symmetric space M can be regarded as a fibre bundle over a Riemannian 2-symmetric space with totally geodesic fibres isometric to a 2-symmetric space. Here the result of R. Crittenden for conjugate and cut points in a 2-symmetric space is extended to the focal points of the fibres of M. Also the restriction of the exponential map of M up to the first focal locus in the normal bundle of a fibre is proved to yield a covering map onto its image. It is shown that for the noncompact dual M*, the fibres have no focal points and hence the exponential map of M* restricted to the normal bundle of a fibre is a covering map. The classification of the compact simply connected 4-symmetric spaces G/L with G classical simple provides a large class of examples of these fibrations.