In questo articolo si riassumono le definizioni e le principali proprietà dei gruppi di ostruzione con decorazione di tipo LS e LP. Si stabiliscono nuove relazioni fra questi gruppi e si descrivono le proprietà delle mappe naturali fra differenti gruppi con decorazione. Si costruiscono varie successioni spettrali, contenenti questi gruppi con decorazione, e si studiano la loro connessione con le successioni spettrali in $K$-teoria per certe estensioni quadratiche di antistrutture. Infine, si introduce il concetto di diagramma geometrico di gruppi e si calcolano esplicitamente i gruppi di ostruzione per un diagramma formato da 2-gruppi finiti.
Referenze Bibliografiche
[1]
P. M. AKHMET'EV,
Splitting homotopy equivalences along a one-sided submanifold of codimension 1,
Izv. Akad. Nauk SSSR Ser. Mat.,
51 (2) (
1987), 211-241 (in Russian); English transl. in
Math. USSR Izv.,
30 (2) (
1988), 185-215. |
Zbl 0643.57016[2]
P. M. AKHMET'EV-
YU. V. MURANOV,
Obstructions to the splitting of manifolds with infinite fundamental group,
Mat. Zametki,
60 (2) (
1996), 163-175 (in Russian); English transl. in,
Math. Notes,
60 (1-2) (
1996), 121-129. |
Zbl 0905.57020[4]
S. BUONCRISTIANO-
C. P. ROURKE-
J. SANDERSON,
A Geometric Approach to Homology Theory,
London Math. Soc. Lect. Note Ser.,
18,
Cambridge Univ. Press, Cambridge-New York-Melbourne,
1976. |
MR 413113 |
Zbl 0315.55002[5]
S. E. CAPPELL-
J. L. SHANESON,
Pseudo-free actions. I., in
Algebraic Topology (Aarhus, 1978),
Lect. Notes in Math.,
763,
Springer-Verlag, Berlin,
1979, 395-447. |
MR 561231 |
Zbl 0416.57020[6]
A. CAVICCHIOLI-
F. HEGENBARTH,
On 4-manifolds with free fundamental group,
Forum Math.,
6 (
1994), 415-429. |
MR 1277705 |
Zbl 0822.57015[7]
A. CAVICCHIOLI-
F. HEGENBARTH,
A note on four-manifolds with free fundamental groups,
J. Math. Sci. Univ. Tokyo,
4 (
1997), 435-451. |
MR 1466355 |
Zbl 0893.57016[8]
A. CAVICCHIOLI-
F. HEGENBARTH-
D. REPOVŠ,
On the stable classification of certain 4-manifolds,
Bull. Austral. Math. Soc.,
52 (
1995), 385-398. |
MR 1358695 |
Zbl 0863.57014[9]
A. CAVICCHIOLI-
F. HEGENBARTH-
D. REPOVŠ,
Four-manifolds with surface fundamental groups,
Trans. Amer. Math. Soc.,
349 (
1997), 4007-4019. |
MR 1376542 |
Zbl 0887.57026[10]
A. CAVICCHIOLI-
YU. V. MURANOV-
D. REPOVŠ,
Spectral sequences in $K$-theory for a twisted quadratic extension,
Yokohama Math. Journal,
46 (
1998), 1-13. |
MR 1670761 |
Zbl 0958.19002[11] A. CAVICCHIOLI-YU. V. MURANOV-D. REPOVŠ, Una introduzione geometrica alla L-teoria, to appear.
[12] R. K. DENNIS - C. PEDRINI - M. R. STEIN (Eds.),
Algebraic $K$-Theory, Commutative Algebra, and Algebraic Geometry, Proceed. U.S.-Italy Joint Sem. (S. Margherita Ligure, June 18-24, 1989),
Contemporary Math.,
126 Amer. Math. Soc. Providence, R.I.,
1992. |
MR 1156497 |
Zbl 0742.00073[13] S. C. FERRY - A. A. RANICKI - J. ROSENBERG (Eds.),
Novikov Conjectures, Index Theorems and Rigidity, Vol. 1,
London Math. Soc. Lecture Notes,
226,
Cambridge Univ. Press, Cambridge,
1995. |
MR 1388294 |
Zbl 0829.00027[14]
M. H. FREEDMAN-
F. QUINN,
Topology of 4-Manifolds,
Princeton Univ. Press, Princeton, N. J.,
1990. |
MR 1201584 |
Zbl 0705.57001[15]
M. H. FREEDMAN-
P. TEICHNER,
4-Manifold topology I: Subexponential groups,
Invent. Math.,
122 (
1995), 509-529. |
MR 1359602 |
Zbl 0857.57017[16]
R. I. GRIGORCHUK,
Degrees of growth of finitely generated groups and the theory of invariant means,
Izv. Akad. Nauk. SSSR Ser. Mat.,
48 (5) (
1984), 939-985 (in Russian); English transl. in
Math. USSR Izvestiya,
25 (
1985), 259-300. |
Zbl 0583.20023[17]
I. HAMBLETON,
Projective surgery obstructions on closed manifolds,
Algebraic $K$K-theory, Part II (Oberwolfach 1980),
Lect. Notes Math. 967,
Springer-Verlag, Berlin (
1982), 101-131. |
MR 689390 |
Zbl 0503.57018[18]
I. HAMBLETON-
A. F. KHARSHILADZE,
A spectral sequence in surgery theory,
Mat. Sb.,
183 (9) (
1992), 3-14 (in Russian); English transl. in,
Russian Acad. Sci. Sb. Math.,
77 (
1994). |
Zbl 0791.57022[19]
I. HAMBLETON-
I. MADSEN,
On the computation of the projective surgery obstruction groups,
K-theory,
7 (
1993), 537-574. |
MR 1268592 |
Zbl 0797.57017[20]
I. HAMBLETON-
YU. V. MURANOV,
Projective splitting obstruction groups for onesided submanifolds,
Mat. Sbornik,
190 (
1999), to appear. |
MR 1740157 |
Zbl 0953.57017[21]
I. HAMBLETON-
A. RANICKI-
L. TAYLOR,
Round $L$-theory,
J. Pure Appl. Algebra,
47 (
1987), 131-154. |
MR 906966 |
Zbl 0638.18003[22]
I. HAMBLETON-
L. TAYLOR-
B. WILLIAMS,
An introduction to maps between surgery obstruction groups (
1984), in
Algebraic Topology (Aarhus, 1982),
Lect. Notes in Math. 1051,
Springer-Verlag, Berlin-New York (
1984), pp. 49-127. |
MR 764576 |
Zbl 0556.57026[23]
I. HAMBLETON-
L. R. TAYLOR-
B. WILLIAMS,
Detection theorems in $K$ and $L$-theory,
J. Pure Appl. Algebra,
63 (
1990), 247-299. |
MR 1047584 |
Zbl 0718.18006[24]
J. A. HILLMAN,
The Algebraic Characterization of Geometric 4-Manifolds,
London Math. Soc. Lect. Note Ser. 198,
Cambridge Univ. Press, Cambridge,
1994. |
MR 1275829 |
Zbl 0812.57001[25]
A. F. KHARSHILADZE,
The generalized Browder-Livesay invariant,
Izv. Akad. Nauk. SSSR: Ser. Mat.,
51 (2) (
1987), 379-401 (in Russian); English transl. in:
Math. USSR Izv.,
30 (2) (
1988), 353-374. |
Zbl 0643.57017[26]
S. LOPEZ DE MEDRANO,
Involutions on Manifolds,
Springer-Verlag, Berlin-Heidelberg-New York,
1971. |
MR 298698 |
Zbl 0214.22501[27]
I. MADSEN-
R. J. MILGRAM,
The Classifying Spaces for Surgery and Cobordism of Manifolds,
Ann. of Math. Studies 92,
Princeton Univ. Press, Princeton, N. J.,
1979. |
MR 548575 |
Zbl 0446.57002[28]
YU. V. MURANOV,
Obstruction groups to splitting and quadratic extensions of antistructures,
Izvestiya RAN: Ser. Mat.,
59 (6) (
1995), 107-132 (in Russian); English transl. in
Izvestiya Math.,
59 (6) (
1995), 1207-1232. |
Zbl 0996.57518[29]
YU. V. MURANOV,
Relative Wall groups and decorations,
Mat. Sbornik,
185 (12) (
1994), 79-100 (in Russian); English transl. in,
Russian Acad. Sci. Sb. Math.,
83 (2) (
1995), 495-514. |
Zbl 0861.57043[30]
YU. V. MURANOV,
Obstructions to surgeries of two-sheeted coverings,
Mat. Sbornik,
131 (3) (
1986), 347-356 (in Russian); English transl. in:
Math. USSR Sbornik,
59 (2) (
1998), 339-348. |
Zbl 0624.57029[31]
YU. V. MURANOV,
Splitting problem,
Trudy MIRAN,
212 (
1996), 123-146 (in Russian); English transl. in
Proc. Steklov Inst. Math.,
212 (
1996), 115-137. |
Zbl 0888.57029[32] YU. V. MURANOV, Projective splitting obstruction groups and geometric antistructures, Abstracts of International Conference Dedicated to 90th Anniversary of L. S. Pontryagin. Geometry and Topology, Moscow, 1998.
[33]
YU. V. MURANOV-
A. F. KHARSHILADZE,
Browder-Livesay groups of abelian 2-groups,
Matem. Sbornik,
181 (8) (
1990), 1061-1098 (in Russian); English transl. in
Math. USSR Sb.,
70 (
1991). |
Zbl 0732.55003[34]
YU. V. MURANOV-
D. REPOVŠ,
Groups of obstructions to surgery and splitting for a manifold pair,
Mat. Sb.,
188 (3) (
1997), 127-142 (in Russian); English transl. in
Russian Acad. Sci. Sb. Math.,
188 (3) (
1997), 449-463. |
Zbl 0881.57038[35]
YU. V. MURANOV-
D. REPOVŠ,
Obstructions to reconstructions from a pair of manifolds,
Uspehi Mat. Nauk.,
51 (4) (
1996), 165-166 (in Russian); English transl. in
Russian Math. Surveys,
51 (4) (
1996), 743-744. |
Zbl 0881.57039[36]
S. P. NOVIKOV,
Algebraic construction and properties of Hermitian analogs of $K$-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and theory of characteristic classes, I, II,
Izv. Akad. Nauk SSSR. Ser. Mat.,
34 (
1970), 253-288 and 475-500 (in Russian); English transl. in
Math. USSR Izv.,
4 (
1970), 257-292 and 479-505. |
Zbl 0233.57009[37]
C. PEDRINI-
C. A. WEIBEL,
$K$-theory and Chow groups on singular varieties, in
Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory I, II (Boulder, Colorado, 1983),
Contemporary Math.,
55 Amer. Math. Soc., Providence, R.I. (
1986), 339-370. |
MR 862641 |
Zbl 0607.14002[38]
A. A. RANICKI,
Exact Sequences in the Algebraic Theory of Surgery,
Math. Notes 26,
Princeton Univ. Press, Princeton, N. J.,
1981. |
MR 620795 |
Zbl 0471.57012[39]
A. A. RANICKI,
The $L$-theory of twisted quadratic extensions,
Canad. J. Math.,
39 (
1987), 345-364. |
MR 899842 |
Zbl 0635.57017[40]
A. A. RANICKI,
Algebraic $L$-theory and Topological Manifolds,
Cambridge Tracts in Mathematics,
Cambridge University Press,
1992. |
MR 1211640 |
Zbl 0767.57002[41]
A. A. RANICKI,
High-dimensional knot theory,
Math. Monograph,
Springer-Verlag, Berlin-Heidelberg-New York,
1998. |
MR 1713074 |
Zbl 0910.57001[42]
R. SWITZER,
Algebraic Topology-Homotopy and Homology,
Grund. Math. Wiss. 212,
Springer-Verlag, Berlin-Heidelberg-New York,
1975. |
MR 385836 |
Zbl 0305.55001[43]
C. T. C. WALL,
Surgery on Compact Manifolds,
Academic Press, London - New York,
1970; Second Edition, A. A. Ranicki, Editor,
Amer. Math. Soc., Providence, R. I.,
1999. |
Zbl 0219.57024[44]
C. T. C. WALL,
On the axiomatic foundations of the theory of Hermitian forms,
Proc. Cambridge Phil. Soc.,
67 (
1970), 243-250. |
MR 251054 |
Zbl 0197.31103[45]
C. T. C. WALL,
Foundations of Algebraic $L$-Theory,
Proc. Conf. Battelle Memorial Inst. (Seattle, WA. 1972),
Lect. Notes Math. 343 Springer-Verlag, Berlin,
1973. |
MR 357550 |
Zbl 0269.18010[47]
C. T. C. WALL,
Classification of Hermitian forms, VI. Group rings,
Ann. of Math. (2),
103 (
1976), 1-80. |
MR 432737 |
Zbl 0328.18006