W. Lück and A. Ranicki :
Chain homotopy projections .
Preprint 73 ,
Sonderforschungsbereich für Geometrie und Analysis, University of Göttingen ,
1986 .
A version of this was published in J. Algebra 120 :2 (1989)
Zbl
0616.57011 techreport
People
BibTeX
@techreport {key0616.57011z,
AUTHOR = {L\"uck, Wolfgang and Ranicki, Andrew},
TITLE = {Chain homotopy projections},
TYPE = {preprint},
NUMBER = {73},
INSTITUTION = {Sonderforschungsbereich f\"ur Geometrie
und Analysis, University of G\"ottingen},
YEAR = {1986},
PAGES = {31},
NOTE = {A version of this was published in \textit{J.
Algebra} \textbf{120}:2 (1989). Zbl:0616.57011.},
}
W. Lück and A. Ranicki :
“Surgery transfer 167–246
in
Algebraic topology and transformation groups
(Göttingen, Germany, 23–29 August 1987 ).
Edited by T. tom Dieck .
Lecture Notes In Mathematics 1361 .
Springer (Berlin ),
1988 .
MR
979509 Zbl
0677.57012 incollection
Abstract
People
BibTeX
Given a Hurewicz fibration \( F \to E \stackrel{p}{\longrightarrow} B \) with fibre an \( n \) -dimensional Poincaré complex \( F \) we construct algebraic transfer maps in the Wall surgery obstruction groups
\[ p^!: L_m(\mathbb{Z}[\pi_1(B)]) \to L_{m+n}(\mathbb{Z}[\pi_1(B)]) \qquad (m\geq 0) \]
and prove that they agree with the geometrically defined transfer maps. In subsequent work we shall obtain specific computations of the composites \( p^!\,p_! \) , \( p_!\,p^! \) with
\[ p_!:L_m(\mathbb{Z}[\pi_1(B)]) \to L_m(\mathbb{Z}[\pi_1(B)]) \]
the change of rings maps, and some vanishing results.
@incollection {key979509m,
AUTHOR = {L\"uck, W. and Ranicki, A.},
TITLE = {Surgery transfer},
BOOKTITLE = {Algebraic topology and transformation
groups},
EDITOR = {tom Dieck, Tammo},
SERIES = {Lecture Notes In Mathematics},
NUMBER = {1361},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1988},
PAGES = {167--246},
DOI = {10.1007/BFb0083036},
NOTE = {(G\"ottingen, Germany, 23--29 August
1987). MR:979509. Zbl:0677.57012.},
ISSN = {0075-8434},
ISBN = {9780387505282},
}
W. Lück and A. Ranicki :
“Chain homotopy projections J. Algebra
120 : 2
(February 1989 ),
pp. 361–391 .
A preprint appeared as Mathematica Göttingensis no. 73 (1986) .
MR
989903 Zbl
0671.18005 article
People
BibTeX
@article {key989903m,
AUTHOR = {L\"uck, Wolfgang and Ranicki, Andrew},
TITLE = {Chain homotopy projections},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {120},
NUMBER = {2},
MONTH = {February},
YEAR = {1989},
PAGES = {361--391},
DOI = {10.1016/0021-8693(89)90202-0},
NOTE = {A preprint appeared as Mathematica G\"ottingensis
no. 73 (1986). MR:989903. Zbl:0671.18005.},
ISSN = {0021-8693},
}
W. Lück and A. Ranicki :
“Surgery obstructions of fibre bundles J. Pure Appl. Algebra
81 : 2
(1992 ),
pp. 139–189 .
MR
1176019 Zbl
0755.57013 article
Abstract
People
BibTeX
In a previous paper we obtained an algebraic description of the transfer maps
\[ p^*: L_n(\mathbb{Z}[\pi_1(B)]) \to L_{n+d}(\mathbb{Z}[\pi_1(E)]) \]
induced in the Wall surgery obstruction groups by a fibration
\[ F \to E \stackrel{p}{\longrightarrow} B \]
with the fibre \( F \) a \( d \) -dimensional Poincaré complex. In this paper we define a \( \pi_1(B) \) -equivariant symmetric signature
\[ \sigma^*(F,\omega) \in L^d(\pi_1(B),\mathbb{Z}) \]
depending only on the fibre transport
\[ \omega: \pi_1(B)\to [F,F] ,\]
and prove that the composite
\[ p_*\,p^*: L_n(\mathbb{Z}[\pi_1(B)]) \to L_{n+d}(\mathbb{Z}[\pi_1(B)]) \]
is the evaluation \( \sigma^*(F,\omega)\,\otimes\,? \) of the product
\[ \otimes: L^d(\pi_1(B),\mathbb{Z})\otimes L_n(\mathbb{Z}[\pi_1(B)]) \to L_{n+d}(\mathbb{Z}[\pi_1(B)]) .\]
This is applied to prove vanishing results for the surgery transfer, such as \( p^* = 0 \) if \( F = G \) is a compact connected \( d \) -dimensional Lie group which is not a torus, and
\[ F \to E \stackrel{p}{\longrightarrow} B \]
is a \( G \) -principal bundle. An appendix relates this expression for \( p^*\,p_* \) to the twisted signature formula of Atiyah, Lusztig and Meyer.
@article {key1176019m,
AUTHOR = {L\"uck, Wolfgang and Ranicki, Andrew},
TITLE = {Surgery obstructions of fibre bundles},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {81},
NUMBER = {2},
YEAR = {1992},
PAGES = {139--189},
DOI = {10.1016/0022-4049(92)90003-X},
NOTE = {MR:1176019. Zbl:0755.57013.},
ISSN = {0022-4049},
}
A. Ranicki :
“Foundations of algebraic surgery ,”
pp. 491–514
in
Topology of high-dimensional manifolds
(Trieste, Italy 21 May–8 June 2001 ).
Edited by F. T. Farrell, L. Göttsche, and W. Lück .
ICTP Lecture Notes 9 .
The Abdus Salam International Centre for Theoretical Physics (Trieste ),
2002 .
MR
1937022 Zbl
1063.57027 ArXiv
math/0111315 incollection
Abstract
People
BibTeX
@incollection {key1937022m,
AUTHOR = {Ranicki, Andrew},
TITLE = {Foundations of algebraic surgery},
BOOKTITLE = {Topology of high-dimensional manifolds},
EDITOR = {Farrell, F. Thomas and G\"ottsche, Lothar
and L\"uck, Wolfgang},
SERIES = {ICTP Lecture Notes},
NUMBER = {9},
PUBLISHER = {The Abdus Salam International Centre
for Theoretical Physics},
ADDRESS = {Trieste},
YEAR = {2002},
PAGES = {491--514},
NOTE = {(Trieste, Italy 21 May--8 June 2001).
ArXiv:math/0111315. MR:1937022. Zbl:1063.57027.},
ISBN = {9295003128},
}
A. Ranicki :
“The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction ,”
pp. 515–538
in
Topology of high-dimensional manifolds
(Trieste, Italy 21 May–8 June 2001 ).
Edited by F. T. Farrell, L. Göttsche, and W. Lück .
ICTP Lecture Notes 9 .
The Abdus Salam International Centre for Theoretical Physics (Trieste, Italy ),
2002 .
MR
1937023 Zbl
1069.57019 ArXiv
math/0111316 incollection
Abstract
People
BibTeX
@incollection {key1937023m,
AUTHOR = {Ranicki, Andrew},
TITLE = {The structure set of an arbitrary space,
the algebraic surgery exact sequence
and the total surgery obstruction},
BOOKTITLE = {Topology of high-dimensional manifolds},
EDITOR = {Farrell, F. Thomas and G\"ottsche, Lothar
and L\"uck, Wolfgang},
SERIES = {ICTP Lecture Notes},
NUMBER = {9},
PUBLISHER = {The Abdus Salam International Centre
for Theoretical Physics},
ADDRESS = {Trieste, Italy},
YEAR = {2002},
PAGES = {515--538},
NOTE = {(Trieste, Italy 21 May--8 June 2001).
ArXiv:math/0111316. MR:1937023. Zbl:1069.57019.},
ISBN = {9295003128},
}
A. Ranicki :
“Circle valued Morse theory and Novikov homology ,”
pp. 539–569
in
Topology of high-dimensional manifolds
(Trieste, Italy, 21 May–8 June 2001 ).
Edited by F. T. Farrell, L. Göttsche, and W. Lück .
ICTP Lecture Notes 9 .
The Abdus Salam International Centre for Theoretical Physics (Trieste, Italy ),
2002 .
MR
1937024 Zbl
1068.57031 ArXiv
math/0111317 incollection
Abstract
People
BibTeX
@incollection {key1937024m,
AUTHOR = {Ranicki, Andrew},
TITLE = {Circle valued {M}orse theory and {N}ovikov
homology},
BOOKTITLE = {Topology of high-dimensional manifolds},
EDITOR = {Farrell, F. Thomas and G\"ottsche, Lothar
and L\"uck, Wolfgang},
SERIES = {ICTP Lecture Notes},
NUMBER = {9},
PUBLISHER = {The Abdus Salam International Centre
for Theoretical Physics},
ADDRESS = {Trieste, Italy},
YEAR = {2002},
PAGES = {539--569},
NOTE = {(Trieste, Italy, 21 May--8 June 2001).
ArXiv:math/0111317. MR:1937024. Zbl:1068.57031.},
ISBN = {9295003128},
}
E. K. Pedersen, F. Quinn, and A. Ranicki :
“Controlled surgery with trivial local fundamental groups 421–426
in
High-dimensional manifold topology
(Trieste, Italy, 21 May–8 June 2001 ).
Edited by F. T. Farrell and W. Lück .
World Scientific (River Edge, NJ ),
2003 .
MR
2048731 Zbl
1050.57025 ArXiv
math/0111269 incollection
Abstract
People
BibTeX
We provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here is based on work of M. Yamasaki.
@incollection {key2048731m,
AUTHOR = {Pedersen, Erik Kj{\ae}r and Quinn, Frank
and Ranicki, Andrew},
TITLE = {Controlled surgery with trivial local
fundamental groups},
BOOKTITLE = {High-dimensional manifold topology},
EDITOR = {Farrell, F. T. and L\"uck, W.},
PUBLISHER = {World Scientific},
ADDRESS = {River Edge, NJ},
YEAR = {2003},
PAGES = {421--426},
DOI = {10.1142/9789812704443_0018},
NOTE = {(Trieste, Italy, 21 May--8 June 2001).
ArXiv:math/0111269. MR:2048731. Zbl:1050.57025.},
ISBN = {9812382232},
}
A. Ranicki :
“Book review: M. Kreck and W. Lück, ‘The Novikov conjecture, geometry and algebra’ Bull. Lond. Math. Soc.
42 : 1
(2010 ),
pp. 181–183 .
MR
2586978 article
People
BibTeX
@article {key2586978m,
AUTHOR = {Ranicki, Andrew},
TITLE = {Book review: {M}. {K}reck and {W}. {L}\"uck,
``The {N}ovikov conjecture, geometry
and algebra''},
JOURNAL = {Bull. Lond. Math. Soc.},
FJOURNAL = {Bulletin of the London Mathematical
Society},
VOLUME = {42},
NUMBER = {1},
YEAR = {2010},
PAGES = {181--183},
DOI = {10.1112/blms/bdp113},
NOTE = {MR:2586978.},
ISSN = {0024-6093},
}
