We construct spectral sequences for computing the cohomology of automorphism groups of formal groups equipped with additional endomorphisms given by a $p$–adic number ring. We then compute the cohomology of the group of automorphisms of a height four formal group law which commute with additional endomorphisms of the group law by the ring of integers in the field ${\mathbb{Q}}_p\left(\sqrt{p}\right)$ for primes . This automorphism group is a large profinite subgroup of the height four strict Morava stabilizer group. The group cohomology of this group of automorphisms turns out to have cohomological dimension $8$ and total rank $80$. We then run the $K\left(4\right)$–local $E_4$–Adams spectral sequence to compute the homotopy groups of the homotopy fixed-point spectrum of this group’s action on the Lubin–Tate/Morava spectrum $E_4$.

Magnitude is a numerical invariant of enriched categories, including in particular metric spacesas $\left[0,\infty \right)$–enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call magnitude homology (in fact, it is a special sort of Hochschild homology), whose graded Euler characteristic is the magnitude. Magnitude homology of metric spaces generalizes the Hepworth–Willerton magnitude homology of graphs, and detects geometric information such as convexity.

Twisted Morava $K$–theory, along with computational techniques, including a universal coefficient theorem and an Atiyah–Hirzebruch spectral sequence, was introduced by Craig Westerland and the first author (J. Topol. 8 (2015) 887–916). We employ these techniques to compute twisted Morava $K$–theory of all connective covers of the stable orthogonal group and stable unitary group, and their classifying spaces, as well as spheres and Eilenberg–Mac Lane spaces. This extends to the twisted case some of the results of Ravenel and Wilson (Amer. J. Math. 102 (1980) 691–748) and Kitchloo, Laures and Wilson (Adv. Math. 189 (2004) 192–236) for Morava $K$–theory. This also generalizes to all chromatic levels computations by Khorami (J. Topol. 4 (2011) 535–542) (and in part those of Douglas in Topology 45 (2006) 955–988) at chromatic level one, ie for the case of twisted $K$–theory. We establish that for natural twists in all cases, there are only two possibilities: either the twisted Morava homology vanishes, or it is isomorphic to untwisted homology. We also provide a variant on the twist of Morava $K$–theory, with mod 2 cohomology in place of integral cohomology.

We study how the cohomology of a type $F_{\infty }$ relatively hyperbolic group pair $\left(G,\mathcal{𝒫}\right)$ changes under Dehn fillings (ie quotients of group pairs). For sufficiently long Dehn fillings where the quotient pair $(\overline{G},\overline{\mathcal{P}})$ is of type $F_{\infty }$, we show that there is a spectral sequence relating the cohomology groups $H^i\left(G,\mathcal{𝒫};\mathbb{Z}G\right)$ and $H^i(\overline{G},\overline{\mathcal{P}};\mathbb{Z}\overline{G})$. As a consequence, we show that essential cohomological dimension does not increase under these Dehn fillings.

We present an array of new calculations in Lagrangian Floer theory which demonstrate observations relating to symplectic reduction, grading periodicity and the closed–open map. We also illustrate Perutz’s symplectic Gysin sequence and the quilt theory of Wehrheim and Woodward.

We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flow; this includes coupled mean curvature/Kähler–Ricci flow in the sense of Smoczyk (Leipzig University, 2001). In particular, we show invariance when a pair of self-intersection points is born or dies at a self-tangency, using results of Ekholm, Etnyre and Sullivan (J. Differential Geom. 71 (2005) 177–305). Using this we prove a lower bound on the time for which the immersed Floer theory is invariant under the flow, if it exists. This proves part of a conjecture of Joyce (EMS Surv. Math. Sci. 2 (2015) 1–62).

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, ie no pair of distinct Dehn surgeries on such a knot produce $3$–manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or $3$–manifolds; for knots, we study the signature and some finite type invariants, and for $3$–manifolds, we deploy the $SL\left(2,ℂ\right)$ Casson invariant.

For a pointed topological space $\mathit{X}$, we use an inductive construction of a simplicial resolution of $\mathit{X}$ by wedges of spheres to construct a “higher homotopy structure” for $\mathit{X}$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $\mathit{X}$ up to weak equivalence. It can also be used to distinguish between different maps $f:\mathit{X}\to \mathit{Y}$ which induce the same morphism $f_{\ast }:{\pi }_{\ast }\mathit{X}\to {\pi }_{\ast }\mathit{Y}$.

Given two $2n$–dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, we show that a certain map from the $n$–torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map on singular homology with mod $2$ coefficients. The proof uses parametrized moduli spaces of $J$–holomorphic cylinders in completed symplectic cobordisms.

A cooriented foliation $\mathcal{ℱ}$ of an oriented $3$–manifold $M$ is taut if and only if there is a map from $M$ to the $2$–sphere whose restriction to every leaf is a branched cover.

In relation to the $4$–dimensional smooth Poincaré conjecture, we construct a tentative invariant of homotopy $4$–spheres using embedded contact homology (ECH) and Seiberg–Witten theory (SWF). But, for good reason, it is a constant value independent of the sphere, so this null result demonstrates that one should not try to use the usual theories of ECH and SWF. On the other hand, a corollary is that there always exist pseudoholomorphic curves satisfying certain constraints in (punctured) $4$–spheres.

The nonorientable $4$–genus ${\gamma }_4\left(K\right)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the $4$–ball and bounding the knot $K$. We study a conjecture proposed by Batson about the value of ${\gamma }_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor’s conjecture for the orientable $4$–genus of torus knots. We prove the conjecture for many infinite families of torus knots, by calculating for all torus knots a lower bound for ${\gamma }_4$ formulated by Ozsváth, Stipsicz and Szabó.

We study existence and lack thereof of closed, embedded, orientable, codimension one, totally geodesic submanifolds of minimal volume, cusped, orientable, hyperbolic manifolds.

Work of Ni and Zhang has shown that, for the torus knot $T_{r,s}$ with , every surgery slope $p∕q\ge \frac{30}{67}\left(r^2-1\right)\left(s^2-1\right)$ is a characterizing slope. We show that this can be lowered to a bound which is linear in $rs$, namely $p∕q\ge \frac{43}{4}\left(rs-r-s\right)$. The main technical ingredient in this improvement is to show that if $Y$ is an $L$–space bounding a sharp $4$–manifold which is obtained by $p∕q$–surgery on a knot $K$ in $S^3$ and $p∕q$ exceeds $4g\left(K\right)+4$, then the Alexander polynomial of $K$ is uniquely determined by $Y$ and $p∕q$. We also show that if $p∕q$–surgery on $K$ bounds a sharp $4$–manifold, then $S_{p^{\prime }∕q^{\prime }}^3\left(K\right)$ bounds a sharp $4$–manifold for all $p^{\prime }∕q^{\prime }\ge p∕q$.