Geodesic
Wall groups of finite groups and $\Pi$-signatures of manifolds
Sbornik. Mathematics, Tome 27 (1975) no. 2, pp. 163-181 Cet article a éte moissonné depuis la source Math-Net.Ru

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This article contains the evaluation of the image of the natural homomorphism $\chi$ from the even-dimensional Wall group $L_{2k}(\Pi)$ into the ring of complex representations of a finite group $\Pi$. The computations are carried out for finite groups acting freely and linearly on spheres, by means of a differential-topological interpretation of $\chi$; the Atiyah–Singer invariant is utilized as a tool. Bibliography: 8 titles. 
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     author = {G. A. Kats},
     title = {Wall groups of finite groups and $\Pi$-signatures of manifolds},
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G. A. Kats. Wall groups of finite groups and $\Pi$-signatures of manifolds. Sbornik. Mathematics, Tome 27 (1975) no. 2, pp. 163-181. http://geodesic.mathdoc.fr/item/SM_1975_27_2_a1/

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[2] T. Petrie, “The Atiyah-Singer invariant, the Wall groups $L_n(\Pi, 1)$ and the function $(te^x+1)/(te^x-1)$”, Ann. Math., 92:1 (1970), 174–187 | DOI | MR | Zbl

[3] C. T. C. Wall, “Some $L$-groups of finite groups”, Bull. Amer. Math. Soc., 79:3 (1973), 526–529 | DOI | MR | Zbl

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[8] J. A. Wolf, “The manifolds covered by a Riemanian homogeneous manifolds”, Amer. Math., 82:4 (1960), 661–688 | DOI | MR | Zbl