On fake Lens spaces with fundamental group of order a power of 2. (English) Zbl 1220.57020

Summary: We present a classification of fake lens spaces of dimension \(\geq 5\) which have as fundamental group the cyclic group of order \(N = 2^K\), which extends the results of Wall and others in the case \(N = 2\).


57R65 Surgery and handlebodies
57S25 Groups acting on specific manifolds
19J25 Surgery obstructions (\(K\)-theoretic aspects)
57R67 Surgery obstructions, Wall groups
Full Text: DOI arXiv


[1] M F Atiyah, I M Singer, The index of elliptic operators. III, Ann. of Math. \((2)\) 87 (1968) 546 · Zbl 0164.24301
[2] P E Conner, E E Floyd, Differentiable periodic maps, Ergebnisse der Math. und ihrer Grenzgebiete 33, Academic Press (1964) · Zbl 0125.40103
[3] I Hambleton, L R Taylor, A guide to the calculation of the surgery obstruction groups for finite groups (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 225 · Zbl 0952.57009
[4] W Lück, A basic introduction to surgery theory (editors F T Farrell, L Göttsche, W Lück), ICTP Lect. Notes 9, Abdus Salam Int. Cent. Theoret. Phys. (2002) 1 · Zbl 1045.57020
[5] T Macko, C Wegner, On the classification of fake lens spaces · Zbl 1236.57044
[6] I Madsen, R J Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Math. Studies 92, Princeton University Press (1979) · Zbl 0446.57002
[7] S López de Medrano, Involutions on manifolds, Ergebnisse der Math. und ihrer Grenzgebiete 59, Springer (1971) · Zbl 0214.22501
[8] J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358 · Zbl 0147.23104
[9] J W Morgan, D P Sullivan, The transversality characteristic class and linking cycles in surgery theory, Ann. of Math. \((2)\) 99 (1974) 463 · Zbl 0295.57008
[10] T Petrie, The Atiyah-Singer invariant, the Wall groups \(L_n(\pi ,\thinspace 1)\), and the function \((te^x+1)/(te^x-1)\), Ann. of Math. \((2)\) 92 (1970) 174 · Zbl 0205.53802
[11] A A Ranicki, A composition formula for manifold structures · Zbl 1193.57011
[12] A A Ranicki, Algebraic \(L\)-theory and topological manifolds, Cambridge Tracts in Math. 102, Cambridge Univ. Press (1992) · Zbl 0767.57002
[13] C T C Wall, Surgery on compact manifolds, Math. Surveys and Monogr. 69, Amer. Math. Soc. (1999) · Zbl 0935.57003
[14] C M Young, Normal invariants of lens spaces, Canad. Math. Bull. 41 (1998) 374 · Zbl 0913.57019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.