Related papers: Real moduli space of stable rational curves revise…

Let $M$ be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group $\mathsf G$ in a way that the quotient space $M/\mathsf G$ has nonempty boundary. Let $\pi : M \to M/\mathsf G$ denote the…

Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,\phi) over X consists of an algebraic vector bundle E over X and a holomorphic section \phi of E. There is a concept of stability for pairs which…

We define Hitchin's moduli space for a principal bundle $P$, whose structure group is a compact semisimple Lie group $K$, over a compact non-orientable Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which identifies…

The rational Chow ring of the moduli space $\mathcal{M}_g$ of curves of genus $g$ is known for $g \leq 6$. Here, we determine the rational Chow rings of $\mathcal{M}_7, \mathcal{M}_8,$ and $\mathcal{M}_9$ by showing they are tautological.…

We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are \'etale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore…

Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for…

In this article we give a survey of homology computations for moduli spaces $\mathfrak{M}_{g,1}^m$ of Riemann surfaces with genus $g\geqslant 0$, one boundary curve, and $m\geqslant 0$ punctures. While rationally and stably this question…

Let $G$ be a Lie group, with an invariant non-degenerate symmetric bilinear form on its Lie algebra, let $\pi$ be the fundamental group of an orientable (real) surface $M$ with a finite number of punctures, and let $\bold C$ be a family of…

We study the topology of the moduli space of flat SU(2)-bundles over a nonorientable surface X. This moduli space may be identified with the space of homomorphisms Hom(\pi_1(X),SU(2)) modulo conjugation by SU(2). In particular, we compute…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N, \partial^h N, \partial^v N)$ with fixed $\partial^v N$, whenever $n \geq 3$ and $(N, \partial^h N)$ is…

In this paper we study spaces of algebras over an operad (non-symmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of…

Let $f$ be a rational map of degree $d\geq 2$. The moduli space $\mathcal{M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of $f$. For $f$ that is not flexible Latt\`es,…

We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie…

Given smooth manifolds $M$ and $N$, manifold calculus studies the space of embeddings $\operatorname{Emb}(M,N)$ via the "embedding tower", which is constructed using the homotopy theory of presheaves on $M$. The same theory allows us to…

If $n \geq 3$, then moduli space ${\mathcal M}_{0,[n+1]}$, of isomorphisms classes of $(n+1)$-marked spheres, is a complex orbifold of dimension $n-2$. Its branch locus ${\mathcal B}_{0,[n+1]}$ consists of the isomorphism classes of those…

We prove that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as that of the ordinary moduli space of curves.

We prove that the rational cohomology $H^i(\mathcal{T}_g;\mathbf{Q})$ of the moduli space of trigonal curves of genus $g$ is independent of $g$ in degree $i<\lfloor g/4\rfloor.$ This makes possible to define the stable cohomology ring as…

We prove, for infinitely many values of $g$ and $n$, the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular, when $n=0$, our results…

The main characters of this paper are the moduli spaces $TM_{g,n}$ of rational tropical curves of genus $g$ with $n$ marked points, with $g\geq 2$. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces…