Cataloguing PL 4-manifolds by gem-complexity

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n=4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices). Possible interactions with the (not completely known) relationship among different classification in TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.Comment: 25 pages, 5 figures. Improvements suggested by the refere

4-colored graphs and knot/link complements

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.Comment: 19 pages, 6 figures, 3 tables; changes in Lemma 6, Corollaries 7 and

The double of the doubles of Klein surfaces

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface $X$ is a Klein surface $Y$ such that there is a degree two morphism (of Klein surfaces) $Y\rightarrow X$. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if $X$ is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: "the double of doubles" denoted by $DX$. We describe how to use the double of doubles in the study of both moduli spaces and automorphisms of Klein surfaces. Furthermore, we show that the morphism from $DX$ to $X$ is not given by the action of an isometry group on classical surfaces.Comment: 14 pages; more details in the proof of theorem

Computing Matveev's complexity via crystallization theory: the boundary case

The notion of Gem-Matveev complexity has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base $\mathbb D^2$ and two exceptional fibers and, therefore, for all torus knot complements.Comment: 27 pages, 14 figure

Planning the adaptation to climate change in cities: An introduction

Trattasi dell'introduzione al libro

Uniform random colored complexes

We present here random distributions on $(D+1)$-edge-colored, bipartite graphs with a fixed number of vertices $2p$. These graphs are dual to $D$-dimensional orientable colored complexes. We investigate the behavior of quantities related to those random graphs, such as their number of connected components or the number of vertices of their dual complexes, as $p \to \infty$. The techniques involved in the study of these quantities also yield a Central Limit Theorem for the genus of a uniform map of order $p$, as $p \to \infty$.Comment: 36 pages, 9 figures, minor additions and correction

TOPOLOGY IN COLORED TENSOR MODELS

From a “geometric topology” point of view, the theory of manifold representation by means of edge-colored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL d-manifold by means of a totally combinatorial tool. Edge-colored graphs also play an important rôle within colored tensor models theory, considered as a possible approach to the study of Quantum Gravity: the key tool is the G-degree of the involved graphs, which drives the 1/N expansion in the higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the G-degree, have specific relevance in the tensor models framework, show- ing a direct fruitful interaction between tensor models and discrete geometry, via edge-colored graphs. In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edge-colored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds - or, at least, singular d-manifolds, if d ≥ 4 - represented by graphs of a given G-degree. In dimension 4, the existence of colored graphs encoding different PL manifolds with the same underlying TOP manifold, suggests also to investigate the ability of ten- sor models to accurately reflect geometric degrees of freedom of Quantum Gravity

Bioindication of troposheric ozone by native vegetation: the potential of Viburnum lantana for large-scale surveys

Tropospheric ozone is one of the most important phytotoxic air pollutant. Biomonitoring by native plants is an option when the assessment of its impact on vegetation in large-scale surveys and in remote areas is of concern. The shrub species Viburnum lantana L. (wayfaring tree) is known to be sensitive to ozone, develops specific symptoms consisting in red stipples on the upper surface, and has a wide spatial distribution. However it is not fully known if it meets all the requirements to be used as an in situ bioindicator (Doley, 2010). In particular, the actual responsiveness to ozone of native plants and the relationship between the intensity of responses and the levels of exposure to the pollutant under field conditions remain to be evaluated. For these purposes, two field studies were carried out in Trentino (North Italy) at local (1) and large scale (2). (1) Firstly, at local level, we considered two 1x1 km quadrates characterized by different ozone levels. A fully randomized design was adopted to ensure within-quadrate replications and to select V. lantana plants for the monitoring of ozone-specific foliar symptoms development, the chlorophyll (Chl) content and the fluorescence of Chl a during the entire growing season. (2) Secondly, a stratified (elevation x ozone) random design was used to test the response of V. lantana in terms of symptomatic plant percentage at larger scale (6200 km2). Overall, the results of the two field studies allowed to verify a temporal development of the responses of V. lantana consistent with the trend of ozone exposure (Gottardini et al., 2010), and an higher frequency of symptomatic plants where ozone levels were also higher. However, the frequency of symptoms was not always proportionate to the level of ozone exposure. At the same time of the onset and spread of foliar symptoms, a decrease in the foliar Chl content and in photosynthetic performance occurred. V. lantana seems suitable as a bioindicator in situ to qualitatively assess the potential impact of ozone