Bordered Heegaard Floer homology and graph manifolds

We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type D trimodule associated to the trivial S^1 bundle over the pair of pants P. The second is a bimodule that is necessary for self-gluing, when two torus boundary components of a bordered manifold are glued to each other. Using the results of these two computations, we describe an algorithm for computing HF-hat of any graph manifold.Comment: 59 pages, 21 figures, new version corrects typos and adds a short discussion of grading

A calculus for bordered Floer homology

We consider a class of manifolds with torus boundary admitting bordered Heegaard Floer homology of a particularly simple form, namely, the type D structure may be described graphically by a disjoint union of loops. We develop a calculus for studying bordered invariants of this form and, in particular, provide a complete description of slopes giving rise to L-space Dehn fillings as well as necessary and sufficient conditions for L-spaces resulting from identifying two such manifolds along their boundaries. As an application, we show that Seifert fibered spaces with torus boundary fall into this class, leading to a proof that, among graph manifolds containing a single JSJ torus, the property of being an L-space is equivalent to non-left-orderability of the fundamental group and to the non-existence of a coorientable taut foliation.Comment: 79 pages, 14 figures, uses tik

Influence of Online Discussion Initial Posts on Peer Engagement in Response Posts

Engagement in online classes can promote student success but can be met with challenges associated with students feeling isolated from their peers, their instructors, or the course when learning online. Building opportunities for student interaction is one way to combat this isolation. A common activity in online courses that allows for student interaction is the asynchronous discussion; however, discussions do not inherently lead to meaningful engagement among students.This study aims to determine how students influence the engagement of their peers, so that instructors can better design student interactions that will promote engagement. Specifically, it investigates how the moves that students make in their initial discussion posts influence the behavioral, social, and cognitive engagement of their peers in response posts. Data were collected from asynchronous online discussions then analyzed to determine how the characteristics of initial posts may predict engagement in peer responses. Characteristics of initial posts included the time from the due date the initial post was made, the initial post word count, its reading ease score, its use of first- or second-person language, and its level of cognitive presence. Results indicate that some characteristics of initial posts do influence the behavioral, social, or cognitive engagement of peers. An initial post’s time from due date influenced peer behavioral engagement in the form of whether a response post was made. In terms of peer social engagement, first-person language, reading ease, and word count were found to influence individual categories or indicators of social presence. Finally, an initial post’s use of first-person language and its cognitive presence level were found to influence cognitive engagement in peers. These results suggest that the actions of individual students can influence the engagement of their peers in online discussion boards. On this basis, the characteristics of initial posts that influenced engagement in peer responses should be taken into consideration in the design of online discussion activities