We show that if a positive integral surgery on a knot $K$ inside a homology sphere $X$ results in an induced knot $K_n\subset X_n\left(K\right)=Y$ which has simple Floer homology then $n\ge 2g\left(K\right)$. Moreover, for $X=S^3$ the three-manifold $Y$ is an $L$–space, and the Heegaard Floer homology groups of $K$ are determined by its Alexander polynomial.

This paper uses results on the classification of minimal triangulations of $3$–manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space $L\left(4k,2k-1\right)$ and the generalised quaternionic space $S^3∕Q_{4k}$ have complexity $k$, where $k\ge 2$. Moreover, it is shown that their minimal triangulations are unique.

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefschetz fibrations in order to describe near-symplectic $4$–manifolds. We first study monodromy representations of higher sides of genus–$1$ simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus–$1$ simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of the total spaces of these fibrations. Our results are generalizations of Kas’ classification theorem of genus–$1$ Lefschetz fibrations, which states that the total space of a nontrivial genus–$1$ Lefschetz fibration over $S^2$ is diffeomorphic to an elliptic surface $E\left(n\right)$ for some $n\ge 1$.

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2\pi }{3}$ but less than $2\pi$. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere $M$ satisfies the following properties:

(1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of $M$;

(2) it embeds isometrically in the 3–dimensional Euclidean space as the boundary of a tetrahedron; and

(3) there is a simple closed geodesic $\gamma$ in $M$ such that a fractional Dehn twist along $\gamma$ converts $M$ to the double of a parallelogram.

We show that every closed, simply connected, spin topological $4$–manifold except $S^4$ and $S^2\times S^2$ admits a homologically trivial, pseudofree, locally linear action of ${\mathbb{Z}}_p$ for any sufficiently large prime number $p$ which is nonsmoothable for any possible smooth structure.

It is well known that very special $\Gamma$–spaces and grouplike $E_{\infty }$–spaces both model connective spectra. Both these models have equivariant analogues in the case when the group acting is finite. Shimakawa defined the category of equivariant $\Gamma$–spaces and showed that special equivariant $\Gamma$–spaces determine positive equivariant spectra. Costenoble and Waner [Trans. Amer. Math. Soc. 326 (1991) 485-505] showed that grouplike equivariant $E_{\infty }$–spaces determine connective equivariant spectra.

We show that with suitable model category structures the category of equivariant $\Gamma$–spaces is Quillen equivalent to the category of equivariant $E_{\infty }$–spaces. We define the units of equivariant ring spectra in terms of equivariant $\Gamma$–spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.

We determine for which $m$ the complete graph $K_m$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups $A_4$, $A_5$ or $S_4$.

We show that for any convex surface $S$ in a contact $3$–manifold, there exists a metric on $S$ and a neighbourhood contact isotopic to $S\times I$ with the contact structure given by $ker\left(udt-\star du\right)$ where $u$ is an eigenfunction of the Laplacian on $S$ and $\star$ is the Hodge star from the metric on $S$. This answers a question posed by Komendarczyk [Trans. Amer. Math. Soc. 358 (2006) 2399–2413].

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki–Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.

We show that for every $n\ge 2$ and any $ϵ>0$ there exists a compact hyperbolic $n$–manifold with a closed geodesic of length less than $ϵ$. When $ϵ$ is sufficiently small these manifolds are non-arithmetic, and they are obtained by a generalised inbreeding construction which was first suggested by Agol for $n=4$. We also show that for $n\ge 3$ the volumes of these manifolds grow at least as $1∕{ϵ}^{n-2}$ when $ϵ\to 0$.

In his work on the Novikov conjecture, Yu introduced Property $A$ as a readily verified criterion implying coarse embeddability. Studied subsequently as a property in its own right, Property $A$ for a discrete group is known to be equivalent to exactness of the reduced group $C^{\ast }$–algebra and to the amenability of the action of the group on its Stone–Čech compactification. In this paper we study exactness for groups acting on a finite dimensional $CAT\left(0\right)$ cube complex. We apply our methods to show that Artin groups of type FC are exact. While many discrete groups are known to be exact the question of whether every Artin group is exact remains open.

We introduce a new construction of a surface link in 4–space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering $T^2$–link is equivalent to the split union of spun $T^2$–links and turned spun $T^2$–links. We show that a certain torus-covering $T^2$–link has a nonclassical link group. We give a certain class of ribbon torus-covering $T^2$–links. We present the quandle cocycle invariant of a certain torus-covering $T^2$–link obtained from a classical braid, by using the quandle cocycle invariants of the closure of the braid.

We endow categories of nonsymmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched nonsymmetric monoidal model categories.

The paper introduces $4$–fold symmetric quandles and $4$–fold symmetric quandle homotopy invariants of $3$–manifolds. We classify $4$–fold symmetric quandles and investigate their properties. When the quandle is finite, we explicitly determine a presentation of its inner automorphism group. We calculate the container of the $4$–fold symmetric quandle homotopy invariant. We also discuss symmetric quandle cocycle invariants and coloring polynomials of $4$–fold symmetric quandles.

We present a method for finding embedded nullhomologous tori in standard $4$–manifolds which can be utilized to change their smooth structure. As an application, we show how to obtain infinite families of simply connected smooth $4$–manifolds with $b^{+}=1$ and $b^{-}=2,\dots ,7$, via surgery on nullhomologous tori embedded in the standard manifolds ${ℂ‘\mathbb{P}}^2\#k{\overline{ℂ‘\mathbb{P}}}^2$, $k=2,\dots ,7$.

Let $BS\left(1,n\right)=\langle a,b\mid ab{a}^{-1}=b^n\rangle$ be the solvable Baumslag–Solitar group, where $n\ge 2$. It is known that $BS\left(1,n\right)$ is isomorphic to the group generated by the two affine maps of the line: $f_0\left(x\right)=x+1$ and $h_0\left(x\right)=nx$. The action on $S^1=ℝ\cup \infty$ generated by these two affine maps $f_0$ and $h_0$ is called the standard affine one. We prove that any faithful representation of $BS\left(1,n\right)$ into ${Diff}^1\left(S^1\right)$ is semiconjugated (up to a finite index subgroup) to the standard affine action.

Let $p$ be a prime greater than three. In the $p$–local stable homotopy groups of spheres, R L Cohen constructed the infinite $\zeta$–element ${\zeta }_{n-1}\in {\pi }_{2p^{n+1}-2p^n+2p-5}\left(S\right)$ of order $p$. In the stable homotopy group ${\pi }_{2p^{n+1}-2p^n+2p^2-3}\left(V\left(1\right)\right)$ of the Smith–Toda spectrum $V\left(1\right)$, X Liu constructed an essential element ${\varpi }_k$ for $k\ge 3$. Let ${\beta }_s^{\ast }=j_0{j}_1{\beta }^s\in {\left[V\left(1\right),S\right]}_{2s{p}^2-2s-2p}$ denote the Spanier–Whitehead dual of the generator ${\beta }_s^{\prime \prime }={\beta }^s{i}_1{i}_0\in {\pi }_{2s{p}^2-2s}\left(V\left(1\right)\right)$, which defines the $\beta$–element ${\beta }_s$. Let ${\xi }_{s,k}={\beta }_{s-1}^{\ast }{\varpi }_k$. In this paper, we show that the composite of R L Cohen’s $\zeta$–element ${\zeta }_{n-1}$ with ${\xi }_{s,n}$ is nontrivial, where $n>4$ and $1<s<p-1$. As a corollary, ${\xi }_{s,n}$ is also nontrivial for $1<s<p-1$.

Consider a group $G$ acting nicely on a simply-connected simplicial complex $X$. Numerous classical methods exist for using this group action to produce a presentation for $G$. For the case that $X∕G$ is $2$–connected, we give a new method that has the novelty that one does not have to identify a fundamental domain for the action. Indeed, the resulting presentation is canonical in the sense that no arbitrary choices need to be made. It can be viewed as a nonabelian analogue of a simple result in the study of equivariant homology.

The hit problem for a cohomology module over the Steenrod algebra $\mathcal{A}$ asks for a minimal set of $\mathcal{A}$–generators for the module. In this paper we consider the symmetric algebras over the field ${\mathbb{F}}_p$, for $p$ an arbitrary prime, and treat the equivalent problem of determining the set of ${\mathcal{A}}^{\ast }$–primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action of the Kudo–Araki–May algebra $\mathcal{K}$, and consider the $\mathcal{K}$–module structure of the primitives, which form a sub $\mathcal{K}$–algebra of the dual of the infinite symmetric algebra. Our examples show that the $\mathcal{K}$–action on the primitives is not free. Our new action encompasses, on the finite symmetric algebras, the operators introduced by Kameko for studying the hit problem.

A gap in [H Rubinstein, M Scharlemann, Genus two Heegaard splittings of orientable 3–manifolds, in “Proceedings of the Kirbyfest”, Geom. Topol. Monogr. 2 (1999) 489–553] is explored: new examples are found of closed orientable 3–manifolds with possibly multiple genus 2 Heegaard splittings. Properties common to all the examples in that paper are not universally shared by the new examples: some of the new examples have Hempel distance 3, and it is not clear that a single stabilization always makes the multiple splittings isotopic.

We show that the flat closing conjecture is true for groups acting properly and cocompactly on a CAT(0) cube complex when the action satisfies the cyclic facing triple property. For instance, this property holds for fundamental groups of 3–manifolds that act freely on CAT(0) cube complexes.

For a given $2r$–th root of unity $\xi$, we give explicit formulas of a family of $3$–variable Laurent polynomials $J_{i,j,k}$ with coefficients in $\mathbb{Z}\left[\xi \right]$ that encode the $6j$–symbols associated with nilpotent representations of $U_{\xi }\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$. For a given abelian group $G$, we use them to produce a state sum invariant ${\tau }^r\left(M,L,h_1,h_2\right)$ of a quadruplet (compact $3$–manifold $M$, link $L$ inside $M$, homology class $h_1\in H_1\left(M,\mathbb{Z}\right)$, homology class $h_2\in H_2\left(M,G\right)$) with values in a ring $R$ related to $G$. The formulas are established by a “skein” calculus as an application of the theory of modified dimensions introduced by the authors and Turaev in [Compos. Math. 145 (2009) 196–212]. For an oriented $3$–manifold $M$, the invariants are related to $\tau \left(M,L,\varphi \in H^1\left(M,{ℂ}^{\ast }\right)\right)$ defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of $U_{\xi }\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$. They refine them as $\tau \left(M,L,\varphi \right)={\sum }_{h_1}{\tau }^r\left(M,L,h_1,\tilde{\varphi }\right)$ where $\tilde{\varphi }$ correspond to $\varphi$ with the isomorphism $H_2\left(M,{ℂ}^{\ast }\right)\simeq H^1\left(M,{ℂ}^{\ast }\right)$.