A locally connected continuum without convergent sequences

We answer a question of Juhasz by constructing under CH an example of a locally connected continuum without nontrivial convergent sequences.Comment: 6 pages. Reprinted from Topology and its Applications, in press, Jan van Mill, A locally connected continuum without convergent sequence

Square compactness and the filter extension property

We show that the consistency strength of κ being 2κ-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juhasz

Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial

We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompressible' and `alternating' with `homogeneous'. We also examine the implications of our proof for alternating links in general.Comment: 37 pages, 28 figures; v2 Main results generalised from alternating links to homogeneous links. Title change

Short proof of a theorem of Juhasz

We give a simple proof of the increasing strengthening of Arhangel'skii's Theorem. Our proof naturally leads to a refinement of this result of Juh\'asz.Comment: 5 page

A survey of Heegaard Floer homology

This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer homology, we list some of its most important applications. Many of these are purely topological results, not referring to Heegaard Floer homology itself. Then, we briefly outline the construction of Lagrangian intersection Floer homology. We construct the Heegaard Floer chain complex as a special case of the above, and try to motivate the role of the various seemingly ad hoc features such as admissibility, the choice of basepoint, and Spin^c-structures. We also discuss the proof of invariance of the homology up to isomorphism under all the choices made, and how to define Heegaard Floer homology using this in a functorial way (naturality). Next, we explain why Heegaard Floer homology is computable, and how it lends itself to the various combinatorial descriptions. The last chapter gives an overview of the definition and applications of sutured Floer homology, which includes sketches of some of the key proofs. Throughout, we have tried to collect some of the important open conjectures in the area. For example, a positive answer to two of these would give a new proof of the Poincar\'e conjecture.Comment: 38 pages, 1 figure, a few minor correction

Concordance maps in knot Floer homology

We show that a decorated knot concordance $C$ from $K$ to $K'$ induces a homomorphism $F_C$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}(S^3) \cong \mathbb{Z}_2$ that agrees with $F_C$ on the $E^1$ page and is the identity on the $E^\infty$ page. It follows that $F_C$ is non-vanishing on $\widehat{HFK}_0(K, \tau(K))$. We also obtain an invariant of slice disks in homology 4-balls bounding $S^3$. If $C$ is invertible, then $F_C$ is injective, hence $\dim \widehat{HFK}_j(K,i) \le \dim \widehat{HFK}_j(K',i)$ for every $i$, $j \in \mathbb{Z}$. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K'$, then $g(K) \le g(K')$, where $g$ denotes the Seifert genus. Furthermore, if $g(K) = g(K')$ and $K'$ is fibred, then so is $K$.Comment: 38 pages, 3 figures, to appear in Geometry and Topolog

A note on discrete sets

We give several partial positive answers to a question of Juhasz and Szentmiklossy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences and their closures with the cardinality of a Hausdorff space, improving known results in the literature.Comment: 14 pages, to appear on Commentationes Mathematicae Universitatis Carolina