arXiv subjects
Publications and source records attributed to Csaba Nagy.
We survey computations and tools concerning the mapping class group of a compact, oriented, connected 3-manifold $M$. We provide a guide to the literature and sketch proofs for various families of irreducible and geometric 3-manifolds. We also consider JSJ and prime decompositions of 3-manifolds, and consequences for their mapping class groups.
Suppose that $f : M \rightarrow B$ is a normal $(q-1)$-smoothing of a $2q$-manifold $M$ over some $(B,\xi)$, where $q$ is even and $B$ is simply-connected. A diffeomorphism of $M$ over $B$ induces an automorphism of the extended intersection form (or ``Q-form") of $f$, which consists of the intersection form of $M$ and the induced homomorphism $H_q(f)$. Assuming that $H_q(B)$ is free, we determine precisely which automorphisms can be realised by such diffeomorphisms. In particular, if $H_{q-1}(B)$ is also free, then we show that every automorphism can be realised. These results are obtained by studying a ``two-sided" version of the extended surgery obstruction, which was introduced in earlier work of the author. As an application, we show that for a complex $q$-dimensional complete intersection, with $q > 2$ even, every automorphism of the cohomology ring is realised by a diffeomorphism.
We characterise the set of fundamental groups for which there exist $n$-manifolds that are $h$-cobordant (hence homotopy equivalent) but not simple homotopy equivalent, when $n$ is sufficiently large. In particular, for $n \ge 12$ even, we show that examples exist for any finitely presented group $G$ such that the involution on the Whitehead group $Wh(G)$ is nontrivial. This expands on previous work, where we constructed the first examples of even-dimensional manifolds that are homotopy equivalent but not simple homotopy equivalent. Our construction is based on doubles of thickenings, and a key ingredient of the proof is a formula for the Whitehead torsion of a homotopy equivalence between such manifolds.
We study quadratic form parameters $Q$ over the integers and extended quadratic forms with values in $Q$, which we call $Q$-forms. Certain form parameters $Q$ appeared in Wall's work on the classification of almost closed $(n-1)$-connected $2n$-manifolds via $Q$-forms. Baues, Ranicki and Schlichting independently developed definitions of extended quadratic forms in more general settings; when restricted to the ring $\mathbb{Z}$, each of those definitions is equivalent to those studied here. In this paper we classify all quadratic form parameters $Q$ over the integers, determine the category of quadratic form parameters $\mathbf{FP}$ and compute the Witt group functor, \[ W_0 \colon \mathbf{FP} \to \mathbf{Ab}, \quad Q \mapsto W_0(Q),\] where $\mathbf{Ab}$ is the category of finitely generated abelian groups and $W_0(Q)$ is the Witt group of nonsingular $Q$-forms.
Kreck proved that two $2q$-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q-1)$-smoothings over the same normal $(q-1)$-type $(B,\xi)$. We show that stable diffeomorphism can be replaced by diffeomorphism if the normal smoothings have isomorphic Q-forms (which consists of the intersection form of the manifold and the induced homomorphism on $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ is free. This proves a special case of Crowley's Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a $(2k-1)$-connected $4k$-manifold with the kernel of a certain bordism map. By the calculations of Senger-Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups. We also obtain, for a simply-connected $4k$-manifold $M$ with normal $(2k-1)$-type $(B,\xi)$ such that $H_{2k}(B)$ is free, an algebraic description of the stable class of $M$, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to $M$. Using this description, we explicitly compute the stable class of manifolds $M$ with rank-$2$ hyperbolic intersection form.
Given a closed $n$-manifold, we consider the set of simple homotopy types of $n$-manifolds within its homotopy type, called its simple homotopy manifold set. We characterise it in terms of algebraic K-theory, the surgery obstruction map, and the homotopy automorphisms of the manifold. We use this to construct the first examples, for all $n \ge 4$ even, of closed $n$-manifolds that are homotopy equivalent but not simple homotopy equivalent. In fact, we construct infinite families of manifolds that are all homotopy equivalent but pairwise not simple homotopy equivalent, and our examples can be taken to be smooth for $n \geq 6$. Our examples are homotopy equivalent to the product of a circle and a lens space. We analyse the simple homotopy manifold sets of these manifolds, determining exactly when they are trivial, finite, or infinite, and investigating their asymptotic behaviour. The proofs involve integral representation theory and class numbers of cyclotomic fields. We also compare with the relation of $h$-cobordism, and produce similar detailed quantitative descriptions of the manifold sets that arise.
The city metaphor for visualizing software systems in 3D has been widely explored and has led to many diverse implementations and approaches. Common among all approaches is a focus on the software artifacts, while the aspects pertaining to the data and information (stored both in databases and files) used by a system are seldom taken into account. We present M3triCity, an interactive web application whose goal is to visualize object-oriented software systems, their evolution, and the way they access data and information. We illustrate how it can be used for program comprehension and evolution analysis of data-intensive software systems. Demo video URL: https://youtu.be/uBMvZFIlWtk
We define a turning of a rank-$2k$ vector bundle $E \to B$ to be a homotopy of bundle automorphisms $\psi_t$ from $\mathbb{Id}_E$, the identity of $E$, to $-\mathbb{Id}_E$, minus the identity, and call a pair $(E, \psi_t)$ a turned bundle. We investigate when vector bundles admit turnings and develop the theory of turnings and their obstructions. In particular, we determine which rank-$2k$ bundles over the $2k$-sphere are turnable. If a bundle is turnable, then it is orientable. In the other direction, complex bundles are turned bundles and for bundles over finite $CW$-complexes with rank in the stable range, Bott's proof of his periodicity theorem shows that a turning of $E$ defines a homotopy class of complex structure on $E$. On the other hand, we give examples of rank-$2k$ bundles over $2k$-dimensional spaces, including the tangent bundles of some $2k$-manifolds, which are turnable but do not admit a complex structure. Hence turned bundles can be viewed as a generalisation of complex bundles. We also generalise the definition of turning to other settings, including other paths of automorphisms, and we relate the generalised turnability of vector bundles to the topology of their gauge groups and the computation of certain Samelson products.
Developers sometimes choose design and implementation shortcuts due to the pressure from tight release schedules. However, shortcuts introduce technical debt that increases as the software evolves. The debt needs to be repaid as fast as possible to minimize its impact on software development and software quality. Sometimes, technical debt is admitted by developers in comments and commit messages. Such debt is known as self-admitted technical debt (SATD). In data-intensive systems, where data manipulation is a critical functionality, the presence of SATD in the data access logic could seriously harm performance and maintainability. Understanding the composition and distribution of the SATDs across software systems and their evolution could provide insights into managing technical debt efficiently. We present a large-scale empirical study on the prevalence, composition, and evolution of SATD in data-intensive systems. We analyzed 83 open-source systems relying on relational databases as well as 19 systems relying on NoSQL databases. We detected SATD in source code comments obtained from different snapshots of the subject systems. To understand the evolution dynamics of SATDs, we conducted a survival analysis. Next, we performed a manual analysis of 361 sample data-access SATDs, investigating the composition of data-access SATDs and the reasons behind their introduction and removal. We identified 15 new SATD categories, out of which 11 are specific to database access operations. We found that most of the data-access SATDs are introduced in the later stages of change history rather than at the beginning. We also observed that bug fixing and refactoring are the main reasons behind the introduction of data-access SATDs.
Code smells indicate software design problems that harm software quality. Data-intensive systems that frequently access databases often suffer from SQL code smells besides the traditional smells. While there have been extensive studies on traditional code smells, recently, there has been a growing interest in SQL code smells. In this paper, we conduct an empirical study to investigate the prevalence and evolution of SQL code smells in open-source, data-intensive systems. We collected 150 projects and examined both traditional and SQL code smells in these projects. Our investigation delivers several important findings. First, SQL code smells are indeed prevalent in data-intensive software systems. Second, SQL code smells have a weak co-occurrence with traditional code smells. Third, SQL code smells have a weaker association with bugs than that of traditional code smells. Fourth, SQL code smells are more likely to be introduced at the beginning of the project lifetime and likely to be left in the code without a fix, compared to traditional code smells. Overall, our results show that SQL code smells are indeed prevalent and persistent in the studied data-intensive software systems. Developers should be aware of these smells and consider detecting and refactoring SQL code smells and traditional code smells separately, using dedicated tools.
We introduce a rational homotopy invariant P of a topological space, which is a quintic tensor on the cohomology. For n > 1, formality of a closed (n-1)-connected manifold of dimension up to 5n-2 is equivalent to vanishing of P and the Bianchi-Massey tensor introduced by Crowley and the second author arXiv:1505.04184. We show also that elements of the group of closed simply-connected spin 8-manifolds with the cohomology of an r-fold connected sum of S^2 x S^6 are determined up to torsion by the value of P.
Code completion is one of the killer features of Integrated Development Environments (IDEs), and researchers have proposed different methods to improve its accuracy. While these techniques are valuable to speed up code writing, they are limited to recommendations related to the next few tokens a developer is likely to type given the current context. In the best case, they can recommend a few APIs that a developer is likely to use next. We present FeaRS, a novel retrieval-based approach that, given the current code a developer is writing in the IDE, can recommend the next complete method (i.e., signature and method body) that the developer is likely to implement. To do this, FeaRS exploits "implementation patterns" (i.e., groups of methods usually implemented within the same task) learned by mining thousands of open source projects. We instantiated our approach to the specific context of Android apps. A large-scale empirical evaluation we performed across more than 20k apps shows encouraging preliminary results, but also highlights future challenges to overcome.
Modern software is developed under considerable time pressure, which implies that developers more often than not have to resort to compromises when it comes to code that is well written and code that just does the job. This has led over the past decades to the concept of "technical debt", a short-term hack that potentially generates long-term maintenance problems. Self-admitted technical debt (SATD) is a particular form of technical debt: developers consciously perform the hack but also document it in the code by adding comments as a reminder (or as an admission of guilt). We focus on a specific type of SATD, namely "On-hold" SATD, in which developers document in their comments the need to halt an implementation task due to conditions outside of their scope of work (e.g., an open issue must be closed before a function can be implemented). We present an approach, based on regular expressions and machine learning, which is able to detect issues referenced in code comments, and to automatically classify the detected instances as either "On-hold" (the issue is referenced to indicate the need to wait for its resolution before completing a task), or as "cross-reference", (the issue is referenced to document the code, for example to explain the rationale behind an implementation choice). Our approach also mines the issue tracker of the projects to check if the On-hold SATD instances are "superfluous" and can be removed (i.e., the referenced issue has been closed, but the SATD is still in the code). Our evaluation confirms that our approach can indeed identify relevant instances of On-hold SATD. We illustrate its usefulness by identifying superfluous On-hold SATD instances in open source projects as confirmed by the original developers.
We prove the "Sullivan Conjecture" on the classification of 4-dimensional complete intersections up to diffeomorphism. Here an $n$-dimensional complete intersection is a smooth complex variety formed by the transverse intersection of $k$ hypersurfaces in $CP^{n+k}$. Previously Kreck and Traving proved the 4-dimensional Sullivan Conjecture when 64 divides the total degree (the product of the degrees of the defining hypersurfaces) and Fang and Klaus proved that the conjecture holds up to the action of the group of homotopy 8-spheres $\Theta_8 = Z/2$. Our proof involves several new ideas, including the use of the Hambleton-Madsen theory of degree-$d$ normal maps, which provide a fresh perspective on the Sullivan Conjecture in all dimensions. This leads to an unexpected connection between the Segal Conjecture for $S^1$ and the Sullivan Conjecture.
We consider branched coverings which are simple in the sense that any point of the target has at most one singular preimage. The cobordism classes of $k$-fold simple branched coverings between $n$-manifolds form an abelian group $Cob^1(n,k)$. Moreover, $Cob^1(*,k) = \bigoplus_{n=0}^{\infty} Cob^1(n,k)$ is a module over $\Omega^{SO}_*$. We construct a universal $k$-fold simple branched covering, and use it to compute this module rationally. As a corollary, we determine the rank of the groups $Cob^1(n,k)$. In the case $n = 2$ we compute the group $Cob^1(2,k)$, give a complete set of invariants and construct generators.
In an $n$-manifold $X$ each element of $H_{n-1}(X; \mathbb{Z}_2)$ can be represented by an embedded codimension-1 submanifold. Hence for any two such submanifolds there is a third one that represents the sum of their homology classes. We construct such a representative explicitly. We describe the analogous construction for codimension-2 co-oriented submanifolds, and examine the special case of oriented and/or co-oriented submanifolds. We also give a lower bound for the number of connected components of the intersection of two oriented codimension-1 submanifolds in terms of the homology classes they represent.
We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the stable homotopy groups of spheres.