The cohomology groups of a closed manifold $M$ can be reconstructed using the gradient flow of a Morse–Smale function $f:M\to ℝ$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. Witten showed that these inequalities can be deduced analytically by studying the asymptotic behavior of the deformed Laplacian operator. Adopting Witten’s approach, we provide an analytic proof for the so-called equivariant Morse inequalities when the underlying manifold is acted upon by the Lie group $\mathbb{𝕋}=S^1$, and the Morse function $f$ is invariant with respect to this action.

We initiate the study of higher-dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$–equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$–CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$–equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$–CW complex. We prove that such a space is unique up to $M$–homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-${\text{F}}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$–equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-${\text{F}}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including ${\text{FP}}_n$, cohomological dimension, and Hochschild cohomological dimension. We also introduce a theory of $M$–equivariant collapsing schemes which gives new results giving sufficient conditions for a monoid to be of type ${\text{F}}_{\infty }$ (or bi-${\text{F}}_{\infty }$). We identify some families of monoids to which these theorems apply, and in particular provide topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left- right- and bi-${\text{FP}}_{\infty }$. This is the first in a series of three papers proving that all one-relator monoids are of type ${\text{FP}}_{\infty }$, settling a question of Kobayashi from 2000.

We consider symplectic fibrations as in Guillemin and Sternberg (J. Funct. Anal. 52 (1983) 106–128) and derive a spectral sequence to compute the Floer cohomology of certain fibered Lagrangians sitting inside a compact symplectic fibration with small monotone fibers and a rational base. We show that if the Floer cohomology with field coefficients of the fibered Lagrangian vanishes, then the Floer cohomology with field coefficients of the total Lagrangian also vanishes. We give an application to certain nontorus fibers of the Gelfand–Cetlin system in flag manifolds, and show that their Floer cohomology vanishes.

Given a Hamiltonian system $(M,\omega ,G,\mu )$, where $(M,\omega )$ is a symplectic manifold and $G$ is a compact connected Lie group acting on $(M,\omega )$ with moment map $\mu :M\to {\mathfrak{𝔤}}^{\ast }$, one may construct the symplectic quotient $(M⁄⁄G,{\omega }_{\text{red}})$, where $M⁄⁄G:={\mu }^{-1}(0)∕G$. Kirwan used the norm-square of the moment map, $|\mu {|}^2$, as a $G$–equivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M⁄⁄G$.

A real Hamiltonian system $(M,\omega ,G,\mu ,\sigma ,\varphi )$ is a Hamiltonian system along with a pair of involutions $(\sigma :M\to M,\varphi :G\to G)$ satisfying certain compatibility conditions. These imply that the fixed-point set $M^{\sigma }$ is a Lagrangian submanifold of $(M,\omega )$ and that $M^{\sigma }⁄⁄G^{\varphi }:=({\mu }^{-1}(0)\cap M^{\sigma })∕G^{\varphi }$ is a Lagrangian submanifold of $(M⁄⁄G,{\omega }_{\text{red}})$. We prove analogues of Kirwan’s theorems that can be used to calculate the ${\mathbb{Z}}_2$–Betti numbers of $M^{\sigma }⁄⁄G^{\varphi }$. In particular, we prove (under appropriate hypotheses) that $|\mu {|}^2$ restricts to a $G^{\varphi }$–equivariantly perfect Morse–Kirwan function on $M^{\sigma }$ over ${\mathbb{Z}}_2$ coefficients, describe its critical set using explicit real Hamiltonian subsystems, prove equivariant formality for $G^{\varphi }$ acting on $M^{\sigma }$, and combine these results to produce formulas for the ${\mathbb{Z}}_2$–Betti numbers of $M^{\sigma }⁄⁄G^{\varphi }$.

We use the projection complex machinery of Bestvina, Bromberg, and Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to trees, then the group is quasiisometric to a finite-dimensional CAT(0) cube complex. We deduce various properties, including the Helly property for hierarchically quasiconvex subsets.

We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of $ℝ$–circles and contains a Hopf link with three components. We also show that the fundamental group of its complement in $S^3$ is not finitely generated. Additionally, we prove that rank-one spherical CR cusps are quotients of horotubes.

We introduce the notion of coupled link cube complexes (CLCCs) as a means of constructing interesting cocompactly cubulated groups. CLCCs are defined locally, making them a useful tool when precise control over the links is required. We study some general properties of CLCCs, such as their (co)homological dimension and criteria for hyperbolicity. Some examples of fundamental groups of CLCCs are RAAGs, RACGs, surface groups and some manifold groups. As immediate applications of our criteria we produce a number of cubulated $3$– and $4$–manifolds with hyperbolic fundamental group.

We study a notion of curvature for finitely generated groups which serves the role of Ricci curvature for Riemannian manifolds. We prove an analog of the Cheeger–Gromoll splitting theorem. As a consequence, we give a geometric characterization of virtually abelian groups. We also explore the relation between this notion of curvature and the growth of groups.

Building on the foundations in our previous paper, we study Segal conditions that are given by finite products, determined by structures we call cartesian patterns. We set up Day convolution on presheaves in this setting and use it to give conditions under which there is a colimit formula for free algebras and other left adjoints. This specializes to give a simple proof of Lurie’s results on operadic left Kan extensions and free algebras for symmetric $\infty$–operads.

It is shown that there exist $Ƶ$–compactifiable manifolds with noncompact boundary which fail to be pseudocollarable.

A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $F{P}_n(\mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG, and that their finitely presentable subgroups are separable.

We construct effective GKM $T^3$–actions with connected stabilizers on the total spaces of the two $S^2$–bundles over $S^6$ with identical GKM graphs. This shows that the GKM graph of a simply connected integer GKM manifold with connected stabilizers does not determine its homotopy type. We complement this by a discussion of the minimality of this example: the homotopy type of integer GKM manifolds with connected stabilizers is indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lower number of fixed points. Regarding geometric structures on the new example, we find an almost complex structure which is invariant under the action of a subtorus. In addition to the minimal example, we provide an analogous example where the torus actions are Hamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integer GKM manifolds.