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Article

Keywords:
fake lens space; higher structure set; $\rho $-invariant; surgery
Summary:
Extending work of many authors we calculate the higher simple structure sets of lens spaces in the sense of surgery theory with the fundamental group of arbitrary order. As a corollary we also obtain a calculation of the simple structure sets of the products of lens spaces and spheres of dimension grater or equal to $3$.
References:
[1] Balko, L’., Macko, T., Niepel, M., Rusin, T.: Higher simple structure sets of lens spaces with the fundamental group of order $2^K$. Topology Appl. 263 (2019), 299–320 (English). MR 3969225
[2] Hambleton, I., Taylor, L.R.: A guide to the calculation of the surgery obstruction groups for finite groups. Surveys on surgery theory, Vol. 1, Ann. of Math. Stud., vol. 145, Princeton Univ. Press, Princeton, NJ, 2000, pp. 225–274. MR MR1747537 (2001e:19007) MR 1747537
[3] López de Medrano, S.: Involutions on manifolds. Springer-Verlag, New York, 1971, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 59. MR MR0298698 (45 #7747) MR 0298698
[4] Macko, T., Wegner, Ch.: On fake lens spaces with fundamental group of order a power of 2. Algebr. Geom. Topol. 9 (2009), no. 3, 1837–1883. MR 2550097 (2010k:57067) DOI:  http://dx.doi.org/10.2140/agt.2009.9.1837 DOI 10.2140/agt.2009.9.1837 | MR 2550097
[5] Macko, T., Wegner, Ch.: On the classification of fake lens spaces. Forum Math. 23 (2011), no. 5, 1053–1091. MR 2836378 DOI:  http://dx.doi.org/10.1515/FORM.2011.038 DOI 10.1515/form.2011.038 | MR 2836378
[6] Madsen, I., Milgram, R.J.: The classifying spaces for surgery and cobordism of manifolds. Annals of Mathematics Studies, vol. 92, Princeton University Press, Princeton, N.J., 1979. MR MR548575 (81b:57014) MR 0548575
[7] Madsen, I., Rothenberg, M.: On the classification of ${G}$-spheres. II. PL automorphism groups. Math. Scand. 64 (1989), no. 2, 161–218. MR 91d:57024 DOI 10.7146/math.scand.a-12253 | MR 1037458
[8] Quinn, F.: A geometric formulation of surgery. Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 500–511. MR 43 #8087 MR 0282375
[9] Wall, C.T.C.: Surgery on compact manifolds. second ed., Mathematical Surveys and Monographs, vol. 69, American Mathematical Society, Providence, RI, 1999, Edited and with a foreword by A. A. Ranicki. MR MR1687388 (2000a:57089) MR 1687388
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