arXiv research

Study automorphisms of pure braid groups on sphere homotopy groups.

Detects free group automorphisms using homology of covers.

Finite type and finitely generated homotopy groups for manifold automorphisms.

Smooth actions of infinite groups linked to homotopy theory.

Study shows automorphism groups of certain hyperbolic manifolds are infinitely generated.

Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.

New examples of manifolds that are homotopy but not simple homotopy equivalent.

Constructing manifold bundles from orbifolds and proving the existence of free subgroups in second homotopy groups.

Researchers simplify the computation of diffeomorphism groups for Morse-Bott foliations.

Let M be the cotangent bundle of S^2, with the standard symplectic structure. By adapting an argument of Gromov we determine the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity. It turns out that S is weakly homotopy equivalent to \Z. π_0(S) is generated by the cla…

We classify the normal CR structures on $S^3$ and their automorphism groups. Together with [3], this closes the classification of normal CR structures on contact 3-manifolds. We give a criterion to compare 2 normal CR structures, and we show that the underlying contact structure is, up to homotopy, unique.

For every finite graph $Γ$, we define a simplicial complex associated to the outer automorphism group of the RAAG $A_Γ$. These complexes are defined as coset complexes of parabolic subgroups of $Out^0(A_Γ)$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotop…

We consider the space $\X$ of Anosov diffeomorphisms homotopic to a fixed automorphism $L$ of an infranilmanifold $M$. We show that if $M$ is the 2-torus $\mathbb T^2$ then $\X$ is homotopy equivalent to $\mathbb T^2$. In contrast, if dimension of $M$ is large enough, we show that $\X$ is rich in homotopy and has infin…

We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homo…

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class …

Study on high-dimensional solid tori reveals infinite generation in their diffeomorphism groups.

Let M be the product of \C P^m and \C P^n, with the standard integral symplectic form. We prove that the inclusion map from the group of symplectic automorphisms of M to its diffeomorphism group is not surjective on homotopy groups. More precisely, it is not surjective on π_j for all odd j \leq \max\{2m-1,2n-1\}. This …

Embeddings of mapping tori for end-periodic graph maps are proven.

Here we study two interesting smooth contractible manifolds, whose boundaries have non-trivial mapping class groups. The first one is a non-Stein contractible manifold, such that every self diffeomorphism of its boundary extends inside; implying that this manifold can not be a loose cork. The second example is a Stein …

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…

The paper defines and explores vector bundles that can be turned and their properties.

Given a principal bundle over a closed manifold, G --> P --> M, let P^{Ad} --> M be the associated adjoint bundle. Gruher and Salvatore showed that the Thom spectrum (P^{Ad})^{-TM} is a ring spectrum whose corresponding product in homology is a Chas-Sullivan type string topology product. We refer to this spectrum as th…

Mapping class group subgroups yield quasi-isometric curve complex.

Each pointed topological space has an associated $π$-module, obtained from action of its first homotopy group on its second homotopy group. For the $3$-ball with a trivial link with $n$-components removed from its interior, its $π$-module $\mathcal{M}_n$ is of free type. In this paper we give an injection of the (exten…

Study Swan modules and homotopy types, resolving Wall and Dyer questions.

Study shows moduli space of fibrations has specific homotopy types.

The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms of homotopy equivalences of a graph fixing a subgraph. This is needed for the sec…

Study of profinite quandles with constructions and characterizations.

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…

Let $Σ$ be a compact oriented surface. The Dehn twist along every simple closed curve $γ\subset Σ$ induces an automorphism of the fundamental group $π$ of $Σ$. There are two possible ways to generalize such automorphisms if the curve $γ$ is allowed to have self-intersections. One way is to consider the `generalized Deh…

We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparam…

The McCool group, denoted $PΣ_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. The cohomology algebra $H^*(PΣ_n, \mathbb{Q})$ was determined by Jensen, McCammond and Meier. We prove that $H^*(PΣ_n, \mathbb{Q})$ is a non-Koszul algebra for $n \geq 4$, which answers a question of Cohen and Pr…

Abstract: Rational decomposition of homeomorphism spaces for manifolds.

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces o…

We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the ou…

Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handl…

We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve that goal, we study the cobordism category of manifolds with positive scalar cu…

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation…

Let $(G,K)$ be a Riemannian symmetric pair of maximal rank, where $G$ is a compact simply connected Lie group and $K$ the fixed point set of an involutive automorphism $σ$. This induces an involutive automorphism $τ$ of the based loop space $Ω(G)$. There exists a maximal torus $T\subset G$ such that the canonical actio…

Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.

Study knot invariants using automorphism groups of free nilpotent groups.

Constructs Cartan geometries from automorphism behaviors.

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. W…

Automorphism groups of Hopf manifolds are finite and have a bounded order.

Automorphisms of pants complex are shown to be inner.

A localisation of the category of n-manifolds is introduced by formally inverting the connected sum construction with a chosen n-manifold Y. On the level of automorphism groups, this leads to the stable diffeomorphism groups of n-manifolds. In dimensions 0 and 2, this is connected to the stable homotopy groups of spher…

Let $M=H_1\cup_S H_2$ be a Heegaard splitting of a closed orientable 3-manifold $M$ (or a bridge decomposition of a link exterior). Consider the subgroup $\mathrm{MCG}^0(H_j)$ of the mapping class group of $H_j$ consisting of mapping classes represented by auto-homeomorphisms of $H_j$ homotopic to the identity, and let…

We consider S^1-families of Legendrian knots in the standard contact R^3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address …