scholarly journals Homology sequence and excision theorem for Euler class group

2011 ◽  
Vol 249 (1) ◽  
pp. 237-254 ◽  
Author(s):  
Yong Yang
Keyword(s):  
Class Group ◽  
Euler Class ◽  
Homology Sequence ◽  
Euler Class Group

scholarly journals Revisiting Nori's question and homotopy invariance of Euler class groups

2010 ◽  
Vol 8 (3) ◽  
pp. 451-480 ◽  
Author(s):  
Mrinal Kanti Das
Keyword(s):  
Class Group ◽  
Euler Class ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Class Groups ◽  
Homotopy Invariance ◽  
Smoothness Assumption ◽  
Polynomial Extension ◽  
Euler Class Group ◽  
Smooth Affine

AbstractThis paper examines the relation between the Euler class group of a Noetherian ring and the Euler class group of its polynomial extension. When the ring is a smooth affine domain, the two groups are canonically isomorphic. This is a consequence of a theorem of Bhatwadekar-Sridharan, which they proved in order to answer a question of Nori on sections of projective modules over such rings. If the smoothness assumption is removed, the result of Bhatwadekar-Sridharan is no longer valid and also the Euler class groups above are not in general isomorphic. In this paper we investigate a variant of Nori's question for arbitrary Noetherian rings and derive several consequences to understand the relation between various groups in the theory of Euler classes.

Presentations and Low-dimensional Homology

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Group Theory ◽  
Simple Proof ◽  
Mapping Class Group ◽  
Class Group ◽  
Euler Class ◽  
Mapping Class ◽  
Basic Invariant ◽  
Homology Groups ◽  
Surface Bundles ◽  
Low Dimensional

This chapter presents explicit computations of the first and second homology groups of the mapping class group. It begins with a simple proof, due to Harer, of the theorem of Mumford, Birman, and Powell; the proof includes the lantern relation, a relation in Mod(S) between seven Dehn twists. It then applies a method from geometric group theory to prove the theorem that Mod(Sɡ) is finitely presentable. It also provides explicit presentations of Mod(Sɡ), including the Wajnryb presentation and the Gervais presentation, and gives a detailed construction of the Euler class, the most basic invariant for surface bundles, as a 2-cocycle for the mapping class group of a punctured surface. The chapter concludes by explaining the Meyer signature cocycle and the important connection of this circle of ideas with the theory of Sɡ-bundles.

Sign in / Sign up

Export Citation Format

Share Document