Abstract
In 1987 Kharshiladze introduced the concept of type for an element in a Wall group, and proved that the elements of first and second type cannot be realized by normal maps of closed manifolds. This approach is sufficiently easy for computing the assembly maps and sometimes very effective. Here we give a geometrical interpretation of this approach by using the Browder–Quinn surgery obstruction groups for filtered manifolds. To understand the obtained relations we give algebraic definitions of the element types which are based on the algebraic surgery theory of Ranicki. Our approach describes a level of indeterminate for the algebraic passing to surgery on a codimension k submanifold of a given Browder–Livesay filtration. Then we study the realization of splitting obstructions by simple homotopy equivalences of closed manifolds, and compute the assembly maps for some classes of groups.
Similar content being viewed by others
References
Akhmetiev P.M., Cavicchioli A., Repovš D.: On realization of splitting obstructions in Browder–Livesay groups for closed manifold pairs. Proc. Edinb. Math. Soc. (2) 43(1), 15–25 (2000)
Bak, A., Muranov, Yu.V.: Splitting along submanifolds, and \({\mathbb L}\) -spectra. Sovrem. Mat. Prilozh. Topol. Anal. Smezh. Vopr. (in Russian) (2003), no. 1, 3–18; English transl. in J. Math. Sci. (N.Y.) 123(4), 4169–4184 (2004)
Bak, A., Muranov, Yu.V.: Normal invariants of manifold pairs and assembly maps, Mat. Sbornik (in Russian) 197(3), 3–24; English transl. in Sbornik Math (2006)
Bak, A., Muranov, Yu.V.: Splitting along a submanifold with filtration, in preparation
Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)
Browder W., Livesay G.R.: Fixed point free involutions on homotopy spheres. Bull. Amer. Math. Soc. 73, 242–245 (1967)
Browder, W., Quinn, F.: A Surgery Theory for G-Manifolds and Stratified Sets, in Manifolds—Tokyo 1973, pp. 27–36. University of Tokyo Press, Tokyo (1975)
Cappell, S.E., Shaneson, J.L.: Pseudo-Free Actions. I. Lecture Notes in Mathematics, vol. 763, pp. 395–447 (1979)
Cavicchioli A., Muranov Yu.V., Spaggiari F.: Relative groups in surgery theory. Bull. Belgian Math. Soc. 12, 109–135 (2005)
Cavicchioli, A., Hegenbarth, F., Muranov, Yu. V., Spaggiari, F.: On the iterated Browder–Livesay invariants (2009, in press)
Hambleton, I.: Projective Surgery Obstructions on Closed Manifolds. Lecture Notes in Mathematics, vol. 967, pp. 101–131 (1982)
Hambleton I., Ranicki A., Taylor L.: Round L-theory. J. Pure Appl. Algebra 47, 131–154 (1987)
Hambleton I., Milgram J., Taylor L., Williams B.: Surgery with finite fundamental group. Proc. Lond. Mat. Soc. 56, 349–379 (1988)
Hambleton, I., Kharshiladze, A.F.: A spectral sequence in surgery theory, Mat. Sbornik (in Russian) 183, 3–14 (1992); English transl. in Russian Acad. Sci. Sb. Math. 77, 1–9 (1994)
Kharshiladze, A.F.: Smooth and piecewise-linear structures on products of projective spaces, Izv. Akad. Nauk SSSR. Ser. Matem. 47(2), 366–383 (1983); English transl. in Math. USSR Izvestiya 22(2), 339–355 (1984)
Kharshiladze, A.F.: Iterated Browder–Livesay invariants and oozing problem, Mat. Zametki (in Russian) 41, 557–563 (1987); English transl. in Math. Notes 41 (1987)
Kharshiladze, A.F.: Surgery on manifolds with finite fundamental groups, Uspechi Mat. Nauk (in Russian) 42, 55–85 (1987); English transl. in Russian Math. Surveys 42 (1987)
Jimenez, R., Muranov, Yu.V., Repovš, D.: Splitting along a submanifold pair, K-theory (2006, in press)
Jimenez R., Muranov Yu.V., Repovš D.: Surgery spectral sequence and manifolds with filtrations.. Trudy MMO 67, 294–325 (2006) (in Russian)
Lopez de Medrano S.: Involutions on Manifolds. Springer, Berlin (1971)
Muranov, Yu.V.: Obstructions to surgery of double coverings, Mat. Sbornik 173, 347–356 (1986); English transl. in Math. USSR Sb. 59 (1988)
Muranov, Yu.V.: Splitting problem, Trudi MIRAN (in Russian) 212, 123–146 (1996); English transl. in Proc. of the Steklov Inst. of Math. 212, 115–137 (1996)
Muranov, Yu.V., Kharshiladze, A.F.: Browder–Livesay groups of abelian 2–groups, Mat. Sbornik 181, 1061–1098 (1990); English transl. in Math. USSR Sb. 70, 499–540 (1991)
Muranov, Yu.V., Repovš, D., Spaggiari, F.: Surgery on triples of manifolds, Mat. Sbornik 8, 139–160 (2003); English transl. in Sbornik Mathematics 194, 1251–1271 (2003)
Ranicki, A.A.: The Total Surgery Obstruction. Lecture Notes in Mathematics, vol. 763, pp. 275–316 (1979)
Ranicki, A.A.: Exact Sequences in the Algebraic Theory of Surgery. Math. Notes, vol. 26. Princeton University Press, Princeton (1981)
Ranicki A.A.: The L-theory of twisted quadratic extensions. Canad. J. Math. 39, 245–364 (1987)
Ranicki, A.A.: Algebraic L-Theory and Topological Manifolds, Cambridge Tracts in Math., vol. 102. Cambridge University Press, Cambridge (1992)
Ranicki, A.A.: Algebraic and Geometric Splittings of the K- and L-groups of Polynomial Extensions. Lecture Notes in Mathematics, vol. 1217, pp. 321–353 (1986)
Switzer, R.: Algebraic Topology–Homotopy and Homology, Grund. Math. Wiss., vol. 212. Springer, Berlin (1975)
Wall, C.T.C.: Surgery on Compact Manifolds. Academic, London (1970); Ranicki, A.A. (ed.) 2nd edn, American Mathematial Society, Providence (1999)
Wall, C.T.C.: Formulae for surgery obstructions. Topology 15, 182–210 (1976); corrigendum ibid. 16, 495–496 (1977)
Wall C.T.C.: Classification of hermitian forms. VI Group rings. Ann. Math. 103, 1–80 (1976)
Weinberger S.: The Topological Classification of Stratified Spaces. The University of Chicago Press, Chicago (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Michor.
Partially supported by the GNSAGA of the National Research Council of Italy, by the MIUR (Ministero della Istruzione, Università e Ricerca) of Italy within the project Proprietà Geometriche delle Varietà Reali e Complesse, and by a Research Grant of the University of Modena and Reggio Emilia (Italy) and of the Universidad Tecnologica de la Mixteca (Mexico). Y. V. Muranov would like to thank the University of Modena and Reggio E. for the hospitality and support.
Rights and permissions
About this article
Cite this article
Cavicchioli, A., Muranov, Y.V. & Spaggiari, F. Assembly maps and realization of splitting obstructions. Monatsh Math 158, 367–391 (2009). https://doi.org/10.1007/s00605-009-0129-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-009-0129-8
Keywords
- Surgery on manifolds
- Assembly map
- Splitting problem
- Browder–Livesay invariants
- Splitting obstruction groups