A Vietoris–Rips complex R is the clique complex of a proximity graph. This graph can arise from a finite set of points in Euclidean n-space, and then there is a natural projection onto its n-dimensional shadow; p : R æ S. An article by Chambers et al.[Cha+10] demonstrates that p induces an isomorphism of fundamental groups for n = 1, 2, while the induced homomorphism fails to be surjective for ...

In this note, we study the symplectic representation of mapping class group. particular, discuss surjecivity restricted to certain classes. It is well-known that itself surjective. fact still surjective after restricting on pseudo-Anosov However, show surjectivity does not hold when orientable pseudo-Anosovs, even reducing its codomain integer matrices with a bi-Perron leading eigenvalue. order...

The theta map sends code polynomials into the ring of Siegel modular forms of even weights. Explicit description of the image is known for g ≤ 3 and the surjectivity of the theta map follows. Instead it is known that this map is not surjective for g ≥ 5. In this paper we discuss the possibility of an embedding between the associated projective varieties. We prove that this is not possible for g...

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of étale topological types under the Galois action. As our main example we show that the surjectivity statement in Grothendieck’s Section Conjecture would follow from the surjectivity of the map from fixed points to continuous homotopy fixed points on the level of connected...

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,ω) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,ω), and assume that the moment map μ : X −→ Lg∗ is proper. We consider the function |μ|2 : X −→ R, and use a version of Morse theory to show that the inclusion map j : μ(0) −→ X induces a surjection j∗ : H∗ G(X) −→ H∗ G(μ−1(0))...