We study modular approximations $Q\left(\ell \right)$, $\ell =3,5$, of the $K\left(2\right)$–local sphere at the prime $2$ that arise from $\ell$–power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with $Q\left(5\right)$ and record Hill, Hopkins and Ravenel’s computation of the homotopy groups of ${TMF}_0\left(5\right)$. Using these tools and formulas of Mahowald and Rezk for $Q\left(3\right)$, we determine the image of Shimomura’s $2$–primary divided $\beta$–family in the Adams–Novikov spectral sequences for $Q\left(3\right)$ and $Q\left(5\right)$. Finally, we use low-dimensional computations of the homotopy of $Q\left(3\right)$ and $Q\left(5\right)$ to explore the rôle of these spectra as approximations to $S_{K\left(2\right)}$.

We construct an infinite collection of knots with the property that any knot in this family has $n$–string essential tangle decompositions for arbitrarily high $n$.

We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken $n$–manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions $n\le 4$. Our main application is to Coxeter groups whose Davis complexes are manifolds; we show that the natural action of these groups on the Davis complex has a hierarchy. Our second result is that the Singer conjecture is equivalent to the cocompact action dimension conjecture, which is a statement about all groups, not just fundamental groups of closed aspherical manifolds.

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly based mapping class groupoid given by the same theory.

To any graph and smooth algebraic curve $C$, one may associate a “hypercurve” arrangement, and one can study the rational homotopy theory of the complement $X$. In the rational case ($C=ℂ$), there is considerable literature on the rational homotopy theory of $X$, and the trigonometric case ($C={ℂ}^{\times }$) is similar in flavor. The case when $C$ is a smooth projective curve of positive genus is more complicated due to the lack of formality of the complement. When the graph is chordal, we use quadratic-linear duality to compute the Malcev Lie algebra and the minimal model of $X$, and we prove that $X$ is rationally $K\left(\pi ,1\right)$.

We show that if $X$ is a piecewise Euclidean $2$–complex with a cocompact isometry group, then every $2$–quasiflat in $X$ is at finite Hausdorff distance from a subset $Q$ which is locally flat outside a compact set, and asymptotically conical.

Let $M$ be a closed, simply connected, smooth manifold. Let ${\mathbb{F}}_p$ be the finite field with $p$ elements, where $p>0$ is a prime integer. Suppose that $M$ is an ${\mathbb{F}}_p$–elliptic space in the sense of Félix, Halperin and Thomas (1991). We prove that if the cohomology algebra $H^{\ast }\left(M,{\mathbb{F}}_p\right)$ cannot be generated (as an algebra) by one element, then any Riemannian metric on $M$ has an infinite number of geometrically distinct closed geodesics. The starting point is a classical theorem of Gromoll and Meyer (1969). The proof uses string homology, in particular the spectral sequence of Cohen, Jones and Yan (2004), the main theorem of McCleary (1987), and the structure theorem for elliptic Hopf algebras over ${\mathbb{F}}_p$ from Félix, Halperin and Thomas (1991).

The category of $\mathfrak{ℭ}$–colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a relative left properness condition, ie that the class of weak equivalences between $\Sigma$–cofibrant operads is closed under cobase change along cofibrations. We also provide an example of Dwyer which shows that the model structure on $\mathfrak{ℭ}$–colored symmetric operads is not left proper.

The goal of this paper is to prove that the classifying spaces of categories of algebras governed by a prop can be determined by using function spaces on the category of props. We first consider a function space of props to define the moduli space of algebra structures over this prop on an object of the base category. Then we mainly prove that this moduli space is the homotopy fiber of a forgetful map of classifying spaces, generalizing to the prop setting a theorem of Rezk.

The crux of our proof lies in the construction of certain universal diagrams in categories of algebras over a prop. We introduce a general method to carry out such constructions in a functorial way.

In Appendix E of Riemannian foliations [Progress in Mathematics 73, Birkhäuser, Boston (1988)], É Ghys proved that any Lie $\mathfrak{g}$–flow is homogeneous if $\mathfrak{g}$ is a nilpotent Lie algebra. In the case where $\mathfrak{g}$ is solvable, we expect any Lie $\mathfrak{g}$–flow to be homogeneous. In this paper, we study this problem in the case where $\mathfrak{g}$ is a $3$–dimensional solvable Lie algebra.

We prove that certain families of Coxeter groups and inclusions $W_1\hookrightarrow W_2\hookrightarrow \cdots$ satisfy homological stability, meaning that in each degree the homology $H_{\ast }\left(B{W}_n\right)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A$, $B$ and $D$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$–action is highly connected. To do this we show that the barycentric subdivision is an instance of the “basic construction”, and then use Davis’s description of the basic construction as an increasing union of chambers to deduce the required connectivity.

In the mid eighties Goldman proved that an embedded closed curve could be isotoped to not intersect a given closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, asked about characterizing simple closed curves algebraically, in terms of the same Lie structure. We show how the Goldman bracket answers these questions for all finite type surfaces. In fact we count self-intersection numbers and mutual intersection numbers for all finite type orientable orbifolds in terms of a new Lie bracket operation, extending Goldman’s. The arguments are purely topological, or based on elementary ideas from hyperbolic geometry.

These results are intended to be used to recognize hyperbolic and Seifert vertices and the gluing graph in the geometrization of three-manifolds. The recognition is based on the structure of the string topology bracket of three-manifolds.

We show that the center of the Goldman Lie algebra associated to a closed orientable surface is generated by the class of the trivial loop. For an orientable nonclosed surface of finite type, the center is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures.

We use the orientation underlying the Hirzebruch genus of level three to map the beta family at the prime $p=2$ into the ring of divided congruences. This procedure, which may be thought of as the elliptic Greek letter beta construction, yields the $f$–invariants of this family.

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions $>8$. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.

We show that symmetric spaces and thick affine buildings which are not of spherical type $A_1^r$ have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT$\left(0\right)$ cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median.

We study the mod-$2$ cohomology spectral sequence arising from delooping the Bousfield–Kan cosimplicial space giving the $2$–nilpotent completion of a connective spectrum $X$. Under good conditions its $E_2$–term is computable as certain nonabelian derived functors evaluated at $H^{\ast }\left(X\right)$ as a module over the Steenrod algebra, and it converges to the cohomology of ${\Omega }^{\infty }X$. We provide general methods for computing the $E_2$–term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples are also considered; in particular, we show that the spectral sequence collapses at $E_2$ when $X$ is a suspension spectrum.

For $X$ a connected finite simplicial complex we consider ${\Delta }^d\left(X,n\right)$, the space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them are equal, and $B^d\left(X,n\right)$, the analogous space of configurations of unordered points. These reduce to the standard configuration spaces of distinct points when $d=1$. We describe the homotopy groups of ${\Delta }^d\left(X,n\right)$ (resp. $B^d\left(X,n\right)$) in terms of the homotopy (resp. homology) groups of $X$ through a range which is generally sharp. It is noteworthy that the fundamental group of the configuration space $B^d\left(X,n\right)$ abelianizes as soon as we allow points to collide, ie $d\ge 2$.

Building on work of Livernet and Richter, we prove that $E_n$–homology and $E_n$–cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore, we show that the associated Yoneda algebra is trivial.

We use an Adams spectral sequence to calculate the $ℝ$–motivic stable homotopy groups after inverting $\eta$. The first step is to apply a Bockstein spectral sequence in order to obtain $h_1$–inverted $ℝ$–motivic $Ext$ groups, which serve as the input to the $\eta$–inverted $ℝ$–motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor–Witt $\left(4k-1\right)$–stem has order $2^{u+1}$, where $u$ is the $2$–adic valuation of $4k$. This answer is reminiscent of the classical image of $J$. We also explore some of the Toda bracket structure of the $\eta$–inverted $ℝ$–motivic stable homotopy groups.

In this paper, we construct new characteristic classes of fiber bundles via flat connections with values in infinite-dimensional Lie algebras of derivations. In fact, choosing a fiberwise metric, we construct a chain map to the de Rham complex on the base space, and show that the induced map on cohomology groups is independent of the choice of metric. Moreover, we show that, applied to a surface bundle, our construction gives Morita–Miller–Mumford classes.

The simple loop conjecture for $3$–manifolds states that every $2$–sided immersion of a closed surface into a $3$–manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the loop theorem to immersed surfaces. We prove the conjecture in the case that the target $3$–manifold admits a geometric structure modeled on $Sol$.