Weakest topology on curves of R2R^2 that makes length a continuous function

We know that the length of curves is not semi-continuous for certain topologies, and is for some others. A natural question I asked myself recently is “what do we need to make length continuous”, i.e.: what is the weakest topology on curves of, let’s say R2, that makes arclength a continuous function ? Can we … Read more

Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I’m a novice in the algebraic topology, so I’m unable to resolve it by myself. Definition Let us call a (pure) n-dimensional polyhedral complex the topological space glued from a finite number of n-dimensional (convex) polyhedra … Read more

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the following generalization: For three given points $a,b,c \in \mathbb{R}^{2}$ define $$A_{\lambda}=\{z\in \mathbb{R}^{2} \;\text{with}\;\; |z-a|+|z-b|+|z-c|=\lambda\}$$ How is the geometric description of … Read more

Points of failure in definition of X- and A-moduli spaces for arbitrary G

In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. What properties of these groups are required for the definition—in particular, what properties of Borel subgroups do Fock and Goncharov assume that, in general, do not … Read more

Homology of Torelli subgroup groups of automorphism groups of free products

For a group G, let G∗n denote the n-fold free product. There is a natural map Aut((Z/kZ)∗n)↦GLn(Z/kZ). Is it known if the group homology of the kernel of this map is finitely generated in each homological degree? This is false if you replace Z/kZ with Z. Answer AttributionSource : Link , Question Author : qqqqqqw … Read more

When is the fundamental group of a fibration a semi-direct product?

Let f:S→B be a fibration from a projective complex surface onto a curve B. Assume that f has no multiple fibres. Then there is an exact sequence π1(F)→π1(S)→π1(B)→1, where F is a generic fiber. If f has a section B→S, then one has a section π1(B)→π1(S), and therefore π1(S) is the semi-direct product of the … Read more

Integral of second fundamental form

Let us have Riemannian manifold M with boundary N. Let F be an immersion, such that F:N→M and B be a second fundamental form on N relative to F. And let f be a function on N. How we can calculate integral ∫NB(grad(f),grad(f))Ω, where Ω is a volume form on N? Is there exist any … Read more

Exponential contraction for the projection on horospheres

A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres (Reference for the geometry of horospheres), namely exponential decay of the projection onto a horosphere. My question is: does a similar exponential decay still holds in a Gromov hyperbolic space ? The precise statement would be something like: Let … Read more

Oppeness on projective morphism when fibers are semi-ample

Let X be projective variety. Consider π:X→B be a projective morphism and take D={b∈B|KXbis semi-ample} Then when D is open and closed ? Answer AttributionSource : Link , Question Author : Larue , Answer Author : Community

On decidability of a homeomorphism with a prescribed pushforward

This is a refinement of my older question A homeomorphism with a prescribed action on the fundamental group – decidable or not? The problem under considreation is the following. Let M,N be two closed manifolds. For the sake of convenience, we assume that they are smooth and of dimension at least 5. We have an … Read more