We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of “semi-algebraic differential forms” in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the de Rham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich’s proof of the formality of the little cubes operad.

We extend the cactus theorem of Dinitz, Karzanov, Lomonosov to metric spaces. In particular we show that if $X$ is a separable continuum which is not separated by $n-1$ points then the set of all $n$–tuples of points separating $X$ can be encoded by an $ℝ$–tree.

The relations of Barratt and Miller are shown to include all relations among the elements $P^i\chi P^{n-i}$ in the mod $p$ Steenrod algebra, and a minimal set of relations is given.

A $\mathcal{Z}$–structure on a group $G$, defined by M Bestvina, is a pair $\left(\widehat{X},Z\right)$ of spaces such that $\widehat{X}$ is a compact ER, $Z$ is a $\mathcal{Z}$–set in $\widehat{X}$, $G$ acts properly and cocompactly on $X=\widehat{X}\setminus Z$ and the collection of translates of any compact set in $X$ forms a null sequence in $\widehat{X}$. It is natural to ask whether a given group admits a $\mathcal{Z}$–structure. In this paper, we show that if two groups each admit a $\mathcal{Z}$–structure, then so do their free and direct products.

On introduit une notion de paire positive de structures de contact sur les variétés de dimension trois qui généralise celle de Eliashberg et Thurston [Confoliations, Univ. Lecture Series 13, Amer. Math. Soc. (1998)] et Mitsumatsu [Ann. Inst. Fourier (Grenoble) 45 (1995) 1407–1421 ; Foliations : geometry and dynamics (Warsaw, 2000) World Sci. Publ., River Edge, NJ (2002) 75–125]. Une telle paire “normale” donne naissance à un champ de plans continu et localement intégrable $\lambda$. On montre que si $\lambda$ est uniquement intégrable et si les structures de contact sont tendues, alors le feuilletage intégral de $\lambda$ est sans composante de Reeb d’âme homologue à zéro. De plus, dans ce cas, la variété ambiante porte un feuilletage sans composante de Reeb. On démontre également un théorème de stabilité “à la Reeb” pour les paires positives de structures tendues.

We introduce the notion of a positive pair of contact structures of a three dimensional manifold which generalises that of Eliashberg, Thurston and Mitsumatsu. A normal such pair gives rise to a continuous, locally integrable plane field $\lambda$. We show that if lambda is uniquely integrable and if the contact structures are tight then the integral foliation of $\lambda$ has no Reeb component whose core is homologous to zero. Moreover, in this case, the ambient manifold carries a foliation without a Reeb component. We also show a Reeb stability theorem for positive pairs of tight structures.

A toric manifold is a compact non-singular toric variety. A torus manifold is an oriented, closed, smooth manifold of dimension $2n$ with an effective action of a compact torus $T^n$ having a non-empty fixed point set. Hence, a torus manifold can be thought of as a generalization of a toric manifold. In the present paper, we focus on a certain class $\mathfrak{M}$ in the family of torus manifolds with codimension one extended actions, and we give a topological classification of $\mathfrak{M}$. As a result, their topological types are completely determined by their cohomology rings and real characteristic classes.

The problem whether the cohomology ring determines the topological type of a toric manifold or not is one of the most interesting open problems in toric topology. One can also ask this problem for the class of torus manifolds. Our results provide a negative answer to this problem for torus manifolds. However, we find a sub-class of torus manifolds with codimension one extended actions which is not in the class of toric manifolds but which is classified by their cohomology rings.

We define a “sutured topological quantum field theory”, motivated by the study of sutured Floer homology of product $3$–manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach to make computations of contact elements in sutured Floer homology over $\mathbb{Z}$ of sutured manifolds $\left(D^2\times S^1,F\times S^1\right)$ where $F$ is finite. This generalises previous results of the author over ${\mathbb{Z}}_2$ coefficients. Our approach elaborates upon the quantum field theoretic aspects of sutured Floer homology, building a noncommutative Fock space, together with a bilinear form deriving from a certain combinatorial partial order; we show that the sutured TQFT of discs is isomorphic to this Fock space.

We prove that an iterated torus knot type in $\left(S^3,{\xi }_{std}\right)$ fails the uniform thickness property (UTP) if and only if it is formed from repeated positive cablings, which is precisely when an iterated torus knot supports the standard contact structure. This is the first complete UTP classification for a large class of knots. We also show that all iterated torus knots that fail the UTP support cabling knot types that are transversely nonsimple.

Let $G$ be a profinite group with finite virtual cohomological dimension and let $X$ be a discrete $G$–spectrum. If $H$ and $K$ are closed subgroups of $G$, with $H◃K$, then, in general, the $K∕H$–spectrum $X^{hH}$ is not known to be a continuous $K∕H$–spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum ${\left(X^{hH}\right)}^{hK∕H}$. To address this situation, we define homotopy fixed points for delta-discrete $G$–spectra and show that the setting of delta-discrete $G$–spectra gives a good framework within which to work. In particular, we show that by using delta-discrete $K∕H$–spectra, there is always an iterated homotopy fixed point spectrum, denoted ${\left(X^{hH}\right)}^{h_{\delta }K∕H}$, and it is just $X^{hK}$.

Additionally, we show that for any delta-discrete $G$–spectrum $Y$, there is an equivalence $\left(\right.Y^{h_{\delta }H}{\left)\right.}^{h_{\delta }K∕H}\simeq Y^{h_{\delta }K}$. Furthermore, if $G$ is an arbitrary profinite group, there is a delta-discrete $G$–spectrum $X_{\delta }$ that is equivalent to $X$ and, though $X^{hH}$ is not even known in general to have a $K∕H$–action, there is always an equivalence ${\left({\left(X_{\delta }\right)}^{h_{\delta }H}\right)}^{h_{\delta }K∕H}\simeq {\left(X_{\delta }\right)}^{h_{\delta }K}.$ Therefore, delta-discrete $L$–spectra, by letting $L$ equal $H,K,$ and $K∕H$, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete $G$–spectra.

The equivariant loop theorem implies the existence of a loop theorem/Dehn’s lemma for $3$–orbifolds that are good (covered by a $3$–manifold). In this note we prove a loop theorem/Dehn’s lemma for any locally orientable $3$–orbifold (good or bad) whose singular set is labeled with powers of 2. The proof is modeled on the standard tower construction.

In this paper, we investigate the behavior of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specifically with the Chas–Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fiber bundles: the fiber bundle ${\Omega }^{n}M\to L^{n}M\to M$ for an $h_{\ast }$–oriented manifold $M$ and the looped fiber bundle $L^{n}F\to L^{n}E\to L^{n}B$ of a fiber bundle $F\to E\to B$ of $h_{\ast }$–oriented manifolds. Our method lies in the construction of Gysin morphisms of spectral sequences. We apply these results to study the ordinary homology of the free loop spaces of sphere bundles and some generalized homologies of the free loop spaces of spheres and projective spaces. For the latter purpose, we construct explicit manifold generators for the homology of these spaces.

We mainly study the derivation algebra of the free Lie algebra and the Chen Lie algebra generated by the abelianization $H$ of a free group, and trace maps. To begin with, we give the irreducible decomposition of the derivation algebra as a $GL\left(n,Q\right)$–module via the Schur–Weyl duality and some tensor product theorems for $GL\left(n,Q\right)$. Using them, we calculate the irreducible decomposition of the images of the Johnson homomorphisms of the automorphism group of a free group and a free metabelian group.

Next, we consider some applications of trace maps: Morita’s trace map and the trace map for the exterior product of $H$. First, we determine the abelianization of the derivation algebra of the Chen Lie algebra as a Lie algebra, and show that the abelianization is given by the degree one part and Morita’s trace maps. Second, we consider twisted cohomology groups of the automorphism group of a free nilpotent group. In particular, we show that the trace map for the exterior product of $H$ defines a nontrivial twisted second cohomology class of it.

For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$ into $J^1\left({ℝ}^2\right)={ℝ}^5$ which lie in pairwise distinct Legendrian isotopy classes and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the Legendrian contact homology DGA does not admit any augmentation over ${\mathbb{Z}}_2$, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in $J^1\left(S^2\right)$ from a similar perspective.

We prove the nonexistence of surjective homomorphisms from knot groups $G\left(8_{21}\right)$, $G\left(9_{12}\right)$, $G\left(9_{24}\right)$, $G\left(9_{39}\right)$ onto $G\left(4_1\right)$ using twisted Alexander polynomials and the numbers of surjective homomorphisms onto $SL\left(2;\mathbb{Z}∕7\mathbb{Z}\right)$.

We study the representation spaces $R\left(K;i\right)$ appearing in Kronheimer and Mrowka’s instanton knot Floer homologies for a class of pretzel knots. In particular, for pretzel knots $P\left(p,q,r\right)$ with $p,q,r$ pairwise coprime, these appear to be nondegenerate and comprise representations in $SU\left(2\right)$ that are not binary dihedral.

We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of sliding circuits can be found using the well-known cyclic sliding operation. This leads to a polynomial time algorithm for deciding the Nielsen–Thurston type of any braid, modulo one well-known conjecture on the speed of convergence of the cyclic sliding operation.

Let ${\mathcal{ℳ}}_g^n$ be the moduli space of Riemann surfaces of genus $g$ with $n$ labeled marked points. We prove that, for $g\ge 2$, the cohomology groups ${\left\{H^i\left({\mathcal{ℳ}}_g^n;\mathbb{Q}\right)\right\}}_{n=1}^{\infty }$ form a sequence of $S_n$–representations which is representation stable in the sense of Church–Farb. In particular this result applied to the trivial $S_n$–representation implies rational “puncture homological stability” for the mapping class group ${\text{ Mod}}_g^n$. We obtain representation stability for sequences ${\left\{H^i\left({\text{ PMod}}^n\left(M\right);\mathbb{Q}\right)\right\}}_{n=1}^{\infty }$, where ${\text{ PMod}}^n\left(M\right)$ is the mapping class group of many connected orientable manifolds $M$ of dimension $d\ge 3$ with centerless fundamental group; and for sequences ${\left\{H^i\left(\right.B{\text{ PDiff}}^n\left(M\right);\mathbb{Q}\left)\right.\right\}}_{n=1}^{\infty }$, where $B{\text{ PDiff}}^n\left(M\right)$ is the classifying space of the subgroup ${\text{ PDiff}}^n\left(M\right)$ of diffeomorphisms of $M$ that fix pointwise $n$ distinguished points in $M$.

Fox has shown that for every closed connected surface smoothly embedded in $S^3$, the closure of each component of its complement is diffeomorphic to the closure of the complement of a handlebody embedded in $S^3$. In this paper, we study a similar “Fox property” for smooth embeddings of $S^p\times S^q\times S^r$ in $S^{p+q+r+1}$.

We prove that, for $p$ an odd prime, every finite $p$–group of rank $3$ acts freely on a finite complex $X$ homotopy equivalent to a product of three spheres.