The Fundamental Crossed Module of the Complement of a Knotted Surface

We prove that if $M$ is a CW-complex and $M^1$ is its 1-skeleton then the crossed module $\Pi_2(M,M^1)$ depends only on the homotopy type of $M$ as a space, up to free products, in the category of crossed modules, with $\Pi_2(D^2,S^1)$. From this it follows that, if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\Pi_2(M,M^1) \to G$ can be re-scaled to a homotopy invariant $I_G(M)$, depending only on the homotopy 2-type of $M$. We describe an algorithm for calculating $\pi_2(M,M^{(1)})$ as a crossed module over $\pi_1(M^{(1)})$, in the case when $M$ is the complement of a knotted surface $\Sigma$ in $S^4$ and $M^{(1)}$ is the handlebody made from the 0- and 1-handles of a handle decomposition of $M$. Here $\Sigma$ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant $I_G$ yields a non-trivial invariant of knotted surfaces in $S^4$ with good properties with regards to explicit calculations.Comment: A perfected version will appear in Transactions of the American Mathematical Societ

Most Cited Articles Published in Brazilian Journals of Economics: Google Scholar Rankings

This paper examines the rankings of the most cited papers published in Brazilian journals of economics since 1990, according to Google Scholar [GS]. Quality research published in Brazil, as measured by academic impact, is mainly done by authors affiliated with international institutions. Half of the articles are co-authored, 7% are authored by graduate students, and women appear to be underrepresented. There is little overlap between authors publishing in domestic journals and authors publishing in international journals. The areas of research more frequently cited by GS and domestic journals are macroeconomics, labor economics, and industrial organization. The most cited articles in international journals are of econometrics, game theory, and development economics. Revista de Economia Política is the top Brazilian journal for the general public, and Revista de Econometria is the top Brazilian journal for academia. GS citations are not a good indicator of journal citationsRankings of Articles, Economists and Departments, Role of Economists.

On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups

We give an interpretation of Yetter's Invariant of manifolds $M$ in terms of the homotopy type of the function space $TOP(M,B(G))$, where $G$ is a crossed module and $B(G)$ is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of $M$, and the weak homotopy type of the crossed module $G$. We use this interpretation to define a twisting of Yetter's Invariant by cohomology classes of crossed modules, defined as cohomology classes of their classifying spaces, in the form of a state sum invariant. In particular, we obtain an extension of the Dijkgraaf-Witten Invariant of manifolds to categorical groups. The straightforward extension to crossed complexes is also considered.Comment: 45 pages. Several improvement

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of $n$ particles in the complex plane, hence to a categorification of the Knizhnik-Zamolodchikov connection. We discuss infinitesimal 2-braidings in a 2-category naturally assigned to every differential crossed module, leading to the notion of a quasi-invariant tensor in a differential crossed module. Finally we prove that quasi-invariant tensors exist in the differential crossed module associated to the String Lie-2-algebra.Comment: v3 - the introduction has been expanded, overall improvements in the presentation. Final version, to appear in Adv. Mat

Categorifying the sl(2,C)sl(2,C) Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra

We construct a flat (and fake-flat) 2-connection in the configuration space of $n$ indistinguishable particles in the complex plane, which categorifies the $sl(2,C)$-Knizhnik-Zamolodchikov connection obtained from the adjoint representation of $sl(2,C)$. This will be done by considering the adjoint categorical representation of the string Lie 2-algebra and the notion of an infinitesimal 2-Yang-Baxter operator in a differential crossed module. Specifically, we find an infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra, proving that any (strict) categorical representation of the string Lie-2-algebra, in a chain-complex of vector spaces, yields a flat and (fake flat) 2-connection in the configuration space, categorifying the $sl(2,C)$-Knizhnik-Zamolodchikov connection. We will give very detailed explanation of all concepts involved, in particular discussing the relevant theory of 2-connections and their two dimensional holonomy, in the specific case of 2-groups derived from chain complexes of vector spaces.Comment: The main result was considerably sharpened. Title, abstract and introduction updated. 50 page

Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.Comment: New results. A perfected version will appear in Compositio Mathematic

The Tenure Game: Building Up Academic Habits

Why do some academics continue to be productive after receiving tenure? This paper answers this question by using a Stackelberg differential game between departments and scholars. We show that departments can set tenure rules and standards as incentives for scholars to accumulate academic habits. As a result, academic habits have a lasting positive impact in scholar’s productivity, leading to higher scholar’s productivity rate of growth and higher productivity level.Role of economists; sociology of economics.