Let $G$ be a discrete group. We give methods to compute, for a generalized (co)homology theory, its values on the Borel construction $EG{\times }_{G}X$ of a proper $G$–CW–complex $X$ satisfying certain finiteness conditions. In particular we give formulas computing the topological $K$–(co)homology $K_{\ast }\left(BG\right)$ and $K^{\ast }\left(BG\right)$ up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups $G$ these formulas are sharp. The main new tools we use for the $K$–theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where $G$ is a finite group these theorems reduce to well-known results of Greenlees and Bökstedt.

Quillen’s famous plus construction plays an important role in many aspects of manifold topology. In our own work [Geometry and Topology 7 (2006) 541–556] on ends of open manifolds, an ability to embed cobordisms provided by the plus construction into the manifolds being studied was a key to completing the main structure theorem. In this paper we develop a “spherical modification” trick that allows for a constructive approach to obtaining those embeddings. More importantly, this approach can be used to obtain more general embedding results. In this paper we develop generalizations of the plus construction (together with the corresponding group-theoretic notions) and show how those cobordisms can be embedded in manifolds satisfying appropriate fundamental group properties. Results obtained here are motivated by, and play an important role in, our ongoing study of noncompact manifolds.

We give homotopy invariant definitions corresponding to three well-known properties of complete intersections, for the ring, the module theory and the endomorphisms of the residue field, and we investigate them for the mod $p$ cochains on a space, showing that suitable versions of the second and third are equivalent and that the first is stronger. We are particularly interested in classifying spaces of groups, and we give a number of examples. The case of rational homotopy theory is treated in [J. Pure Appl. Algebra 217 (2013) 636–663], and there are some interesting contrasts.

If $F$ is a compact surface with boundary, then a finitely generated subgroup without peripheral elements of $G={\pi }_1\left(F\right)$ can be separated from finitely many other elements of $G$ by a finite index subgroup of $G$ corresponding to a finite cover $\tilde{F}$ with the same number of boundary components as $F$.

We employ the Goodwillie spectral sequence for the iterated loop space functor in order to provide realizability conditions on certain unstable modules over the Steenrod algebra at an odd prime.

The generalized character map of Hopkins, Kuhn, and Ravenel [J. Amer. Math. Soc. 13 (2000) 553–594] can be interpreted as a map of cohomology theories beginning with a height $n$ cohomology theory $E$ and landing in a height $0$ cohomology theory with a rational algebra of coefficients that is constructed out of $E$. We use the language of $p$–divisible groups to construct extensions of the generalized character map for Morava $E$–theory to every height between $0$ and $n$.

Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic $3$–manifolds. We prove, by a constructive argument, that every veering triangulation admits positive angle structures, recovering a result of Hodgson, Rubinstein, Segerman, and Tillmann. Our construction leads to explicit lower bounds on the smallest angle in this positive angle structure, and to information about angled holonomy of the boundary tori.

Let $f:M\to ℝ$ be a Morse–Bott function on a finite-dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j\to ℝ$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades [Int. Math. Res. Not. 42 (2004) 2179–2269]. Similar data, which also includes a parameter $\epsilon >0$ that scales the Morse-Smale functions $f_j$, can be used to define an explicit perturbation of the Morse-Bott function $f$ to a Morse-Smale function $h_{\epsilon }:M\to ℝ$ [Progr. Math. 133 (1995) 123–183; Ergodic Theory Dynam. Systems 29 (2009) 1693–1703]. In this paper we show that the Morse–Smale–Witten chain complex of $h_{\epsilon }$ is the same as the Morse chain complex defined using cascades for any $\epsilon >0$ sufficiently small. That is, the two chain complexes have the same generators, and their boundary operators are the same (up to a choice of sign). Thus, the Morse Homology Theorem implies that the homology of the cascade chain complex of $f:M\to ℝ$ is isomorphic to the singular homology $H_{\ast }\left(M;\mathbb{Z}\right)$.

We establish a vanishing result for indices of certain twisted Dirac operators on ${Spin}^c$–manifolds with nonabelian Lie group actions. We apply this result to study nonabelian symmetries of quasitoric manifolds. We give upper bounds for the degree of symmetry of these manifolds.

We develop some aspects of the theory of derivators, pointed derivators and stable derivators. Stable derivators are shown to canonically take values in triangulated categories. Similarly, the functors belonging to a stable derivator are canonically exact so that stable derivators are an enhancement of triangulated categories. We also establish a similar result for additive derivators in the context of pretriangulated categories. Along the way, we simplify the notion of a pointed derivator, reformulate the base change axiom and give a new proof that a combinatorial model category has an underlying derivator.

By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum and Vogel and using a construction of Ohtsuki, we define a module endomorphism for each knot $K$ by using a tangle obtained from a surgery presentation of $K$. We show that it is strong shift equivalent to the Turaev–Viro endomorphism associated to $K$. Following Viro, we consider the endomorphisms that one obtains after coloring the meridian and the longitude of the knot. We show that the traces of these endomorphisms encode the same information as the colored Jones polynomials of $K$ at a root of unity. Most of the discussion is carried out in the more general setting of infinite cyclic covers of $3$–manifolds.

Derived $A_{\infty }$–algebras were developed recently by Sagave. Their advantage over classical $A_{\infty }$–algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived $A_{\infty }$–algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad $d\mathcal{A}s$ encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad $A_{\infty }$ as a resolution of the operad $\mathcal{A}s$ encoding associative algebras. We further show that Sagave’s definition of morphisms agrees with the infinity-morphisms of $d{A}_{\infty }$–algebras arising from operadic machinery. We also study the operadic homology of derived $A_{\infty }$–algebras.

Lattice cohomology, defined by Némethi in [Publ. Res. Inst. Math. Sci. 44 (2008) 507–543], is an invariant of negative definite plumbed $3$–manifolds which conjecturally computes their Heegaard Floer homology $H{F}^{+}$. We prove a surgery exact triangle for the lattice cohomology analogous to the one for $H{F}^{+}$. This is a step towards relating these two invariants.

In a well known work, Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps through a range of dimensions increasing with degree. In this paper, we address if a similar result holds when other (not necessarily integrable) almost complex structures are put on projective space. We take almost complex structures that are compatible with the underlying symplectic structure. We obtain the following result: the inclusion of the space of based degree–$k$ $J$–holomorphic maps from ${\mathbb{P}}^1$ to ${\mathbb{P}}^2$ into the double loop space of ${\mathbb{P}}^2$ is a homology surjection for dimensions $j\le 3k-3$. The proof involves constructing a gluing map analytically in a way similar to McDuff and Salamon, and Sikorav, and then comparing it to a combinatorial gluing map studied by Cohen, Cohen, Mann, and Milgram.

Suppose $K$ is a hyperbolic knot in a solid torus $V$ intersecting a meridian disk $D$ twice. We will show that if $K$ is not the Whitehead knot and the frontier of a regular neighborhood of $K\cup D$ is incompressible in the knot exterior, then $K$ admits at most one exceptional surgery, which must be toroidal. Embedding $V$ in $S^3$ gives infinitely many knots $K_n$ with a slope $r_n$ corresponding to a slope $r$ of $K$ in $V$. If $r$ surgery on $K$ in $V$ is toroidal then either $K_n\left(r_n\right)$ are toroidal for all but at most three $n$, or they are all atoroidal and nonhyperbolic. These will be used to classify exceptional surgeries on wrapped Montesinos knots in a solid torus, obtained by connecting the top endpoints of a Montesinos tangle to the bottom endpoints by two arcs wrapping around the solid torus.

The goal of this paper is twofold. First, given a Seifert surface $R$ in the $3$–sphere, we show how to construct a Heegaard diagram for the sutured manifold $S^3\left(R\right)$ complementary to $R$, which in turn enables us to compute the sutured Floer homology of $S^3\left(R\right)$ combinatorially. Secondly, we outline how the sutured Floer homology of $S^3\left(R\right)$, together with the Seifert form of $R$, can be used to decide whether two minimal genus Seifert surfaces of a given knot are isotopic in $S^3$. We illustrate our techniques by showing that the knot $8_3$ has two minimal genus Seifert surfaces up to isotopy. Furthermore, for any $n\ge 1$ we exhibit a knot $K_n$ that has at least $n$ nonisotopic free minimal genus Seifert surfaces.

We give a complete description of the closure of the space of one-generator closed subgroups of ${PSL}_2\left(ℝ\right)$ for the Chabauty topology, by computing explicitly the matrices associated with elements of $Aut\left(\mathbb{D}\right)\cong {PSL}_2\left(ℝ\right)$, and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps ${\phi }_n:X\to Y$ transform convergent sequences of closed subsets of the domain $X$ into convergent sequences of closed subsets of the range $Y$. In particular, this allows us to compute certain geometric limits of ${PSL}_2\left(ℝ\right)$ only by looking at the Hausdorff limit of some closed subsets of $ℂ$.

We prove that the universal enveloping algebra of the Lawrence–Sullivan construction is a particular perturbation of the complete Baues–Lemaire cylinder of $S^0$. Together with other evidence we present, this exhibits the Lawrence–Sullivan construction as the right model for $I^{+}$. From this, we also deduce a generalized Euler formula on Bernoulli numbers.

We characterize free products admitting a faithful and highly transitive action. In particular, we show that the group ${PSL}_2\left(\mathbb{Z}\right)\simeq \left(\mathbb{Z}∕2\mathbb{Z}\right)\ast \left(\mathbb{Z}∕3\mathbb{Z}\right)$ admits a faithful and highly transitive action on a countable set.

For a knot $K$ in $S^3$, the ${sl}_2$–colored Jones function $J_K\left(n\right)$ is a sequence of Laurent polynomials in the variable $t$ that is known to satisfy non-trivial linear recurrence relations. The operator corresponding to the minimal linear recurrence relation is called the recurrence polynomial of $K$. The AJ Conjecture (see Garoufalidis [Proceedings of the Casson Fest (2004) 291–309]) states that when reducing $t=-1$, the recurrence polynomial is essentially equal to the $A$–polynomial of $K$. In this paper we consider a stronger version of the AJ Conjecture, proposed by Sikora [arxiv:0807.0943], and confirm it for all torus knots.