On the Topology of Simple Fold Maps

“…For results about simple fold maps, see, for example, [18,30,31,32,33,39,50]. For results about τ -cobordisms of fold maps with other set τ of singular fibers, see [20,38].…”

Section: Introductionmentioning

confidence: 99%

Cobordism invariants of fold maps

Kalmar¹

2008

Real and Complex Singularities

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Abstract. This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for −1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallelizable manifolds. In the last section some of these results are extended and we show that our invariants also detect stable homotopy groups of the classifying spaces BO(k) as direct summands of the cobordism semigroups of fold maps. IntroductionFold maps of (n + q)-dimensional manifolds into n-dimensional manifolds have the formulaas a local form around each singular point, and the subset of the singular points in the source manifold is a (q + 1)-codimensional submanifold (for results about fold maps, see, for example, [1,2,3,5,8,9,18,26,35,36]). If we restrict a fold map to the set of its singular points, then we obtain a codimension one immersion into the target manifold of the fold map. This immersion together with more detailed informations about the neighbourhood of the set of singular points in the source manifold can be used as a geometrical invariant (see Section 2) of fold cobordism classes (see Definition 1.1) of fold maps (for results about cobordisms of singular maps with completely different approach from our present paper, see, for example, [7,11,12,21,25,34,48] and the works of Ando, Sadykov, Szűcs and the author in References). In this way we obtain a geometrical relation between fold maps and immersions with prescribed normal bundles via cobordisms. In [18] we showed that these invariants describe completely the cobordisms of simple fold maps of (n + 1)-dimensional manifolds into n-dimensional manifolds and in [17] we showed that these invariants detect direct summands of the cobordism group of fold maps, namely stable homotopy groups of spheres. In this paper we extend the results of [17] and show by constructing fibrations of Morse functions over immersed manifolds 2000 Mathematics Subject Classification. Primary 57R45; Secondary 57R75, 57R42, 55Q45.

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“…Generally in toxicity studies, the study director can make a decision on it based on the preliminary studies. Sakuma [44] suggested selecting a one-or two-sided test based on the reports on similar studies conducted. In screening tests for new drugs, the two-sided test is recommended.…”

Section: Which Test To Be Used One-sided or Two-sided?mentioning

confidence: 99%

Transition of Japan’s statistical tools by decision tree for quantitative data obtained from the general repeated dose administration toxicity studies in rodents

Kobayashi¹,

Pillai²,

Michael³

et al. 2014

IJBAS

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Statistical significance is one of important criteria on judgment of regulatory toxicological testing. The decision tree for analysing quantitative data obtained from repeated dose administration studies in rodents has been in use in Japan around 1981. Since then, several authors proposed improved versions of the decision tree incorporating all possible situations of statistical analysis normally encountered in such studies. Recently, a decision tree, which traces a simple route, unlike the previously proposed ones which trace complex routes has been proposed by a few researchers in Japan. While tracing to the most appropriate statistical tool using a decision tree, we propose to consider following points which also play a significant role in selecting the most appropriate statistical tool: (1) statistical tools that fails to detect a significant difference in the low dose group, (2) use of the one-sided test with high power to detect a significant difference compared with two-sided, (3) as far as possible avoid carrying out statistical analysis on the transformed data, since the analytical result of such data is difficult to interpret, (4) it is important to mention what statistical tools of the decision tree are used for the analysis, (5) examine the data for both normality and homogeneity and (6) for testing homogeneity, use Levene's test. Selection of widely accepted statistical tools is usually preferred to less popular and complex statistical analysis. It has been observed that in recent years the preferred statistical tools for analyzing quantitative data obtained from toxicity studied are of simple in nature but with high power to detect a significant difference.

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On the Topology of Simple Fold Maps

Cited by 31 publications

References 7 publications

Remarks on the Topology of Folds

Remarks on the Topology of Folds

Cobordism invariants of fold maps

Transition of Japan’s statistical tools by decision tree for quantitative data obtained from the general repeated dose administration toxicity studies in rodents

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