Surgery on Homology Manifolds-II: The Surgery Exact Sequence and an Application to Fake Tori

1976 ◽  
Vol s2-12 (2) ◽  
pp. 169-175 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Exact Sequence ◽  
Surgery Exact Sequence ◽  
Homology Manifolds

scholarly journals Mapping the surgery exact sequence for topological manifolds to analysis

2017 ◽  
Vol 09 (02) ◽  
pp. 329-361 ◽  
Author(s):  
Vito Felice Zenobi
Keyword(s):  
Exact Sequence ◽  
Dirac Operator ◽  
Index Theorem ◽  
Fundamental Result ◽  
Natural Mapping ◽  
Signature Operator ◽  
Surgery Exact Sequence ◽  
Topological Manifolds ◽  
Stability Results ◽  
Odd Dimensions

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of Higson and Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.

scholarly journals The surgery exact sequence, K-theory and the signature operator

2016 ◽  
Vol 1 (2) ◽  
pp. 109-154 ◽  
Author(s):  
Paolo Piazza ◽  
Thomas Schick
Keyword(s):  
Exact Sequence ◽  
Signature Operator ◽  
Surgery Exact Sequence ◽  
K Theory

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

scholarly journals Topology of homology manifolds

1993 ◽  
Vol 28 (2) ◽  
pp. 324-329 ◽  
Author(s):  
J. Bryant ◽  
S. Ferry ◽  
W. Mio ◽  
S. Weinberger
Keyword(s):  
Homology Manifolds

WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA

2004 ◽  
Vol 15 (10) ◽  
pp. 987-1005 ◽  
Author(s):  
MAHMOUD BENKHALIFA
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Integral Domain ◽  
Free Lie Algebra ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Equivalent Class

Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

Resolutions of Homology Manifolds: A Classification Theorem

1975 ◽  
Vol s2-11 (4) ◽  
pp. 474-480 ◽  
Author(s):  
Allan L. Edmonds ◽  
Ronald J. Stern
Keyword(s):  
Classification Theorem ◽  
Homology Manifolds

scholarly journals A Birman exact sequence for the Torelli subgroup of Aut(Fn)

2016 ◽  
Vol 26 (03) ◽  
pp. 585-617 ◽  
Author(s):  
Matthew Day ◽  
Andrew Putman
Keyword(s):  
Exact Sequence ◽  
Recent Work ◽  
Homology Group ◽  
Entire Group ◽  
Torelli Group

We develop an analogue of the Birman exact sequence for the Torelli subgroup of [Formula: see text]. This builds on earlier work of the authors, who studied an analogue of the Birman exact sequence for the entire group [Formula: see text]. These results play an important role in the authors’ recent work on the second homology group of the Torelli group.

A Characterization of Homology Manifolds

1977 ◽  
Vol s2-16 (1) ◽  
pp. 149-159 ◽  
Author(s):  
Clint McCrory
Keyword(s):  
Homology Manifolds
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