*Andrew Ranicki*

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic ...
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This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.Less

This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.

*Stanley Chang and Shmuel Weinberger*

- Published in print:
- 2019
- Published Online:
- May 2021
- ISBN:
- 9780691160498
- eISBN:
- 9780691200354
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160498.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses some of the techniques that enter into L theory. The nature of L groups for groups π is very different if π is finite or if π is torsion-free. The latter case seems to be ...
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This chapter discusses some of the techniques that enter into L theory. The nature of L groups for groups π is very different if π is finite or if π is torsion-free. The latter case seems to be governed by rigidity phenomena and is conjecturally very closely related to group homology. Finite groups, however, are studied algebraically. The chapter then explains some of the mathematics that surround the calculation of L groups, looking at connections to Witt theory, algebraic K theory, and representation theory. The connection to representation theory motivates Dress induction. The chapter considers Shaneson's thesis which shows that the Farrell fibering theorem is equivalent to a calculation of L groups.Less

This chapter discusses some of the techniques that enter into *L* theory. The nature of *L* groups for groups *π* is very different if *π* is finite or if *π* is torsion-free. The latter case seems to be governed by rigidity phenomena and is conjecturally very closely related to group homology. Finite groups, however, are studied algebraically. The chapter then explains some of the mathematics that surround the calculation of *L* groups, looking at connections to Witt theory, algebraic *K* theory, and representation theory. The connection to representation theory motivates Dress induction. The chapter considers Shaneson's thesis which shows that the Farrell fibering theorem is equivalent to a calculation of *L* groups.