Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Mukherjee, Amiya (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

Gromov formulated and proved a very general Smale - Hirsch - Phillips type theorem. The theorem concerns the claseification of a class of cross-sections o of a smooth fibre bundle E X over a non-closed manifold X such that each r-jet j(0) satisfies an openness condition as well es a stability condition. Briefly, the openness condition is that each j(a) is a cross-section of some preacribed open subbundle of the bundle of r-jets of local oross- sections of EX, and. the etability condition is that this subbundle remain invarient under the action of the paeudogroup of local diffeo- morphisme of x.A model ofgenerative trans formational grammar has proved to be a most efficient and revealing analytical device in the study of the Başdi verbal systen. The model utilised here is one modified from Noam Chomsky s Aspects of the Theory of Syntax and from developments within this model. This frame- vork has proved most appropriate because of its primary concern with syntax and because its generative base provides a set of explicit rules which account for the complex morphology.In that many verb constituents and their inter-relationships can be delineated within the transformational component, the des- cription of the most important parts of the verb is consider-s- in R (both oriented) is in a 1-1 correspondence with the set of integers z. During 1958-59, Smale 1291, [301, t311 clerified the Whitney-Graustein theorem and proved his femous theorem i the set of based regular homotopy classes of based immersions of s into R& n< m, corrosponds bijectively with the n th Stiefel manifold of n-fremes in R . Here a based immersion f is an immersion such that both r and df take prescribed values at a given bese point. A more geometric and conceptual proof of the Smale theorem was found by Thom (33 1. Hirech C161 generalized the Smele theorem for arbitrary menifold pairs proving that, if dim X < dim Y, the correspondence fdf sets up a bijection between the peth components of the spece Imm (X,Y) and the path componente of the space Mono (TX, TY). A further generalization of the immersion theorem states that the differential map d Imm (x,Y) Mono (TX,TY) is, in fact, a woak homotopy aquivalence (u.h.e.). This theorem was afirst proved by Hirsch and Palais (17 1. A proof of the theorem moy be found in Poénaru, [25 1 or (261, Por the differentiable oase, and in Haefliger and Poáneru [14 1 for the combinatoriel case.The technique, mainly geometric, as it evolved with these theorens lay essentially in ahowing certain mapa to be Serre fibra- tions. In the next important exposition of this technique Phillipa [23 ) proved an analogoue result for submersions if dim X > dim Y and X ie non-closed, then the differential map d : Sub-(X,Y) Epi (TX,TY) is a w.h.e.


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