Solving four-dimensional surgery problems using controlled theory
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 4, pp. 221-236.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, the controlled surgery sequence of Ranicki, Pedersen, and Quinn is applied to the solution of surgery problems in dimension four when the fundamental group is not known to be good. Our examples concern free non-Abelian fundamental groups, surface fundamental groups, and special knot groups. Using results from our earlier paper (joint with Spaggiari), we state a general result from which our examples follow.
@article{FPM_2005_11_4_a17,
     author = {F. Hegenbarth and D. Repov\v{s}},
     title = {Solving four-dimensional surgery problems using controlled theory},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {221--236},
     publisher = {mathdoc},
     volume = {11},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2005_11_4_a17/}
}
TY  - JOUR
AU  - F. Hegenbarth
AU  - D. Repovš
TI  - Solving four-dimensional surgery problems using controlled theory
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2005
SP  - 221
EP  - 236
VL  - 11
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2005_11_4_a17/
LA  - ru
ID  - FPM_2005_11_4_a17
ER  - 
%0 Journal Article
%A F. Hegenbarth
%A D. Repovš
%T Solving four-dimensional surgery problems using controlled theory
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2005
%P 221-236
%V 11
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2005_11_4_a17/
%G ru
%F FPM_2005_11_4_a17
F. Hegenbarth; D. Repovš. Solving four-dimensional surgery problems using controlled theory. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 4, pp. 221-236. http://geodesic.mathdoc.fr/item/FPM_2005_11_4_a17/

[1] Cappell S., “Mayer–Vietoris sequences in Hermitian K-theory”, Batelle Inst. Conf. 1972, Lect. Notes. Math., 343, Springer, New York, 478–507 | MR

[2] Cavicchioli A., Hegenbarth F., “On $4$-manifolds with free fundamental groups”, Forum Math., 6 (1994), 415–429 | DOI | MR | Zbl

[3] Cavicchioli A., Hegenbarth F., Repovš D., “Four-manifolds with surface fundamental groups”, Trans. Amer. Math. Soc., 349 (1997), 4007–4019 | DOI | MR | Zbl

[4] Cohen M., “Simplicial structures and transverse cellularity”, Ann. of Math., 85 (1967), 218–245 | DOI | MR | Zbl

[5] Davermann R. J., Decomposition of Manifolds, Academic Press, Orlando, 1986 | MR

[6] Ferry S., Epsilon-delta surgery over $\mathbb Z$, Preprint, 2003 | MR

[7] Freedman M. H., “The topology of four-dimensional manifolds”, J. Differential Geom., 17 (1982), 357–453 | MR | Zbl

[8] Freedman M. H., Quinn F. S., Topology of $4$-Manifolds, Princeton Univ. Press, Princeton, 1990 | MR | Zbl

[9] Freedman M., Teichner P., “$4$-manifold topology. I. Subexponential groups”, Invent. Math., 122 (1995), 509–529 | DOI | MR | Zbl

[10] Freedman M., Teichner P., “$4$-manifold topology. II. Dwyer's fibration and surgery kernels”, Invent. Math., 122 (1995), 531–557 | DOI | MR | Zbl

[11] Hegenbarth F., Repovš D., Spaggiari F., “Connected sums of $4$-manifolds”, Topology Appl. (to appear) | DOI | MR

[12] Jones L., “Patchspaces: A geometric representation for Poincaré spaces”, Ann. of Math., 97 (1973), 276–306 ; Corrigendum, ibid, 102 (1975), 183–185 | DOI | MR | MR | Zbl

[13] Krushkal V., Lee R., “Surgery on closed manifolds with free fundamental groups”, Math. Proc. Cambridge Philos. Soc., 133 (2002), 305–310 | DOI | MR | Zbl

[14] Krushkal V., Quinn F., “Subexponential groups in $4$-manifold topology”, Geom. Topol., 4 (2000), 407–430 | DOI | MR | Zbl

[15] Nicas A., Induction theorems for groups of homotopy manifold structures, Mem. Amer. Math. Soc., 267, 1982 | MR | Zbl

[16] Pedersen E. K., Quinn F., Ranicki A., “Controlled surgery with trivial local fundamental groups”, Proc. School on High-Dimensional Manifold Topology (ICTP, Trieste, 2001), eds. T. Farrell, W. Lück, Word Sci. Press, Singapore, 2003, 421–426 | MR | Zbl

[17] Quinn F., “A geometric formulation of surgery”, Topology of Manifolds, Proc. 1969 Georgia Topology Conference, Markham Press, Chicago, 1970, 500–511 | MR

[18] Quinn F., “Controlled and low-dimensional topology, and a theorem of Keldysh”, Abstracts of Int. Conf. on Geometric Topology, Discrete Geometry, and Set Theory, dedicated to the centennary of L. V. Keldysh (Moscow, August 24–28, 2004), Steklov Mathematical Institute, Moscow, 2004, 8

[19] Ranicki A., “The algebraic theory of surgery. II. Applications to topology”, Proc. London Math. Soc. (3), 40 (1980), 193–283 | DOI | MR | Zbl

[20] Ranicki A. A., Algebraic $L$-Theory and Topological Manifolds, Cambridge Univ. Press, Cambridge, 1992 | MR

[21] Wall C. T. C., Surgery on Compact Manifolds, Academic Press, New York, 1971 | MR | Zbl

[22] West J., “Mapping Hilbert cube manifolds to ANR's: A solution to a conjecture of Borsuk”, Ann. of Math., 106 (1977), 1–18 | DOI | MR | Zbl

[23] Yamasaki M., “$L$-groups of crystallographic groups”, Invent. Math., 88 (1987), 571–602 | DOI | MR | Zbl

[24] Yamasaki M., Relatively simple chain complexes, Preprint, 2004 | MR