Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures, which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of $1+1=2$ on an abacus. The Wheels conjecture is proved from the fact that the $k$–fold connected cover of the unknot is the unknot for all $k$.

Along the way, we find a formula for the invariant of the general $\left(k,l\right)$ cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo–Kirillov map $S\left(\mathfrak{g}\right)\to U\left(\mathfrak{g}\right)$ for metrized Lie (super-)algebras $\mathfrak{g}$.

We characterize which cobounded quasigeodesics in the Teichmüller space $\mathcal{T}$ of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path $\gamma$ in $\mathcal{T}$, we show that $\gamma$ is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over $\gamma$ is a hyperbolic metric space. As an application, for complete hyperbolic 3–manifolds $N$ with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically infinite ends of $N$, a key step in Minsky’s proof of Thurston’s ending lamination conjecture for such manifolds.

We construct a canonical Thom isomorphism in Grojnowski’s equivariant elliptic cohomology, for virtual $\mathbb{T}$–oriented $\mathbb{T}$–equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga’s weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow.

Let $p$ be an odd regular prime, and assume that the Lichtenbaum–Quillen conjecture holds for $K\left(\mathbb{Z}\left[1∕p\right]\right)$ at $p$. Then the $p$–primary homotopy type of the smooth Whitehead spectrum $Wh\left(\ast \right)$ is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted $S^1$-transfer map $t:\Sigma ℂP^{\infty }\to S$. The homotopy groups of $Wh\left(\ast \right)$ are determined in a range of degrees, and the cohomology of $Wh\left(\ast \right)$ is expressed as an $A$-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

We calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres. These calculations can be used to determine also these groups for other three-manifolds, including the product of a circle with a genus two surface.

In an earlier paper, we introduced a knot invariant for a null-homologous knot $K$ in an oriented three-manifold $Y$, which is closely related to the Heegaard Floer homology of $Y$. In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann’s famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with compact boundary, the condition of inward tameness has substatial implications for the algebraic topology at infinity. In particular, every inward tame manifold with compact boundary has stable homology (in all dimensions) and semistable fundamental group at each of its ends. In contrast, we also construct examples of this sort which fail to have perfectly semistable fundamental group at infinity. In doing so, we exhibit the first known examples of open manifolds that are inward tame and have vanishing Wall finiteness obstruction at infinity, but are not pseudo-collarable.

We show that there are algorithms to determine if a 3–manifold contains an essential lamination or a Reebless foliation.

We prove that, if $M$ is a compact oriented manifold of dimension $4k+3$, where $k>0$, such that ${\pi }_1\left(M\right)$ is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to $M$ but not homeomorphic to it. To show the infinite size of the structure set of $M$, we construct a secondary invariant ${\tau }_{\left(2\right)}:S\left(M\right)\to ℝ$ that coincides with the $\rho$–invariant of Cheeger–Gromov. In particular, our result shows that the $\rho$–invariant is not a homotopy invariant for the manifolds in question.

The deformation space of a simplicial $G$–tree $T$ is the set of $G$–trees which can be obtained from $T$ by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as $T$. We give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an $Aut\left(G\right)$–invariant $G$–tree. This gives a sufficient condition for a JSJ splitting to be invariant under automorphisms of $G$. More precisely, the theorem claims that a deformation space contains at most one strongly slide-free $G$–tree, where strongly slide-free means the following: whenever two edges $e_1,e_2$ incident on a same vertex $v$ are such that $G_{e_1}\subset G_{e_2}$, then $e_1$ and $e_2$ are in the same orbit under $G_v$.

We show that a free period three action on the three-sphere is standard, ie, the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts.

A 3–manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3–manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture.

First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3–manifolds. We took the complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3–manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem.

Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3–manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the figure-8 knot is virtually Haken.

The group $\Gamma$ of automorphisms of the polynomial

$$\kappa \left(x,y,z\right)=x^2+y^2+z^2-xyz-2$$

is isomorphic to

$$PGL\left(2,\mathbb{Z}\right)⋉\left(\mathbb{Z}∕2\oplus \mathbb{Z}∕2\right).$$

For $t\in ℝ$, the $\Gamma$–action on ${\kappa }^{-1}\left(t\right)\cap ℝ$ displays rich and varied dynamics. The action of $\Gamma$ preserves a Poisson structure defining a $\Gamma$–invariant area form on each ${\kappa }^{-1}\left(t\right)\cap ℝ$. For $t<2$, the action of $\Gamma$ is properly discontinuous on the four contractible components of ${\kappa }^{-1}\left(t\right)\cap ℝ$ and ergodic on the compact component (which is empty if $t<-2$). The contractible components correspond to Teichmüller spaces of (possibly singular) hyperbolic structures on a torus $\overline{M}$. For $t=2$, the level set ${\kappa }^{-1}\left(t\right)\cap ℝ$ consists of characters of reducible representations and comprises two ergodic components corresponding to actions of $GL\left(2,\mathbb{Z}\right)$ on ${\left(ℝ∕\mathbb{Z}\right)}^2$ and ${ℝ}^2$ respectively. For $2<t\le 18$, the action of $\Gamma$ on ${\kappa }^{-1}\left(t\right)\cap ℝ$ is ergodic. Corresponding to the Fricke space of a three-holed sphere is a $\Gamma$–invariant open subset $\Omega \subset {ℝ}^3$ whose components are permuted freely by a subgroup of index $6$ in $\Gamma$. The level set ${\kappa }^{-1}\left(t\right)\cap ℝ$ intersects $\Omega$ if and only if $t>18$, in which case the $\Gamma$–action on the complement $\left({\kappa }^{-1}\left(t\right)\cap ℝ\right)-\Omega$ is ergodic.

Consider a sequence of pointed $n$–dimensional complete Riemannian manifolds $\left\{\left(M_i,g_i\left(t\right),O_i\right)\right\}$ such that $t\in \left[0,T\right]$ are solutions to the Ricci flow and $g_i\left(t\right)$ have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an $n$–dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov–Hausdorff sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.

Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a combinatorial compactification due to Penner of a space closely related to Riemann’s moduli space. Algebras over these operads are shown to be Batalin–Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.