Splitting obstructions and properties of objects in the Nil categories

Tadeusz Koźniewski

Fundamenta Mathematicae (1999)

  • Volume: 161, Issue: 1-2, page 155-165
  • ISSN: 0016-2736

Abstract

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We show that the objects of Bass-Farrell categories which represent 0 in the corresponding Nil groups are precisely those which are stably triangular. This extends to Waldhausen's Nil group of the amalgamated free product with index 2 factors. Applications include a description of Cappell's special UNil group and reformulations of those splitting and fibering theorems which use the Nil groups.

How to cite

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Koźniewski, Tadeusz. "Splitting obstructions and properties of objects in the Nil categories." Fundamenta Mathematicae 161.1-2 (1999): 155-165. <http://eudml.org/doc/212397>.

@article{Koźniewski1999,
abstract = {We show that the objects of Bass-Farrell categories which represent 0 in the corresponding Nil groups are precisely those which are stably triangular. This extends to Waldhausen's Nil group of the amalgamated free product with index 2 factors. Applications include a description of Cappell's special UNil group and reformulations of those splitting and fibering theorems which use the Nil groups.},
author = {Koźniewski, Tadeusz},
journal = {Fundamenta Mathematicae},
keywords = {triangular object; Bass-Farrell categories; Nil-groups; -groups; -groups; splitting; fibering},
language = {eng},
number = {1-2},
pages = {155-165},
title = {Splitting obstructions and properties of objects in the Nil categories},
url = {http://eudml.org/doc/212397},
volume = {161},
year = {1999},
}

TY - JOUR
AU - Koźniewski, Tadeusz
TI - Splitting obstructions and properties of objects in the Nil categories
JO - Fundamenta Mathematicae
PY - 1999
VL - 161
IS - 1-2
SP - 155
EP - 165
AB - We show that the objects of Bass-Farrell categories which represent 0 in the corresponding Nil groups are precisely those which are stably triangular. This extends to Waldhausen's Nil group of the amalgamated free product with index 2 factors. Applications include a description of Cappell's special UNil group and reformulations of those splitting and fibering theorems which use the Nil groups.
LA - eng
KW - triangular object; Bass-Farrell categories; Nil-groups; -groups; -groups; splitting; fibering
UR - http://eudml.org/doc/212397
ER -

References

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  1. [Bd] B. Badzioch, K 1 of twisted polynomial rings, K-Theory 16 (1999), 29-34. 
  2. [Bs] H. Bass, Algebraic K-Theory, Benjamin, New York, 1968. 
  3. [C1] S. Cappell, Unitary nilpotent groups and Hermitian K-theory, Bull. Amer. Math. Soc. 80 (1974), 1117-1122. Zbl0322.57020
  4. [C2] S. Cappell, Manifolds with fundamental group a generalized free product, ibid. 80 (1974), 1193-1198. Zbl0341.57007
  5. [C3] S. Cappell, A splitting theorem for manifolds, Invent. Math. 33 (1976), 69-170. Zbl0348.57017
  6. [CK] F. Connolly and T. Koźniewski, Nil groups in K-theory and surgery theory, Forum Math. 7 (1995), 45-76. Zbl0844.57036
  7. [F] F. T. Farrell, The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J. 21 (1971), 315-346. Zbl0242.57016
  8. [FH1] F. T. Farrell and W. C. Hsiang, A formula for K 1 R α [ T ] , in: Proc. Sympos. Pure Math. 17, Amer. Math. Soc., 1970, 192-218. 
  9. [FH2] F. T. Farrell and W. C. Hsiang, Manifolds with π 1 = G × α T , Amer. J. Math. 95 (1973), 813-848. 
  10. [KS] S. Kwasik and R. Schultz, Unitary nilpotent groups and the stability of pseudoisotopies, Duke Math. J. 71 (1993), 871-887. Zbl0804.57009
  11. [R] A. Ranicki, Lower K- and L-Theory, Cambridge Univ. Press, 1992. 
  12. [W1] F. Waldhausen, Whitehead groups of generalized free products, preprint, 1969. 
  13. [W2] F. Waldhausen, Algebraic K-theory of generalized free products, Ann. of Math. 108 (1978), 135-256. Zbl0407.18009

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