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Every group is the group of self-homotopy equivalences of finite dimensional $\mathrm{CW}$-complex
Sbornik. Mathematics, Tome 215 (2024) no. 9, pp. 1182-1201 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove that any group $G$ occurs as $\mathcal{E}(X)$, where $X$ is a $\mathrm{CW}$-complex of finite dimension and $\mathcal{E}(X)$ denotes its group of self-homotopy equivalences. Thus, we generalize a well-known theorem due to Costoya and Viruel [9] asserting that any finite group occurs as $\mathcal{E}(X)$, where $X$ is rational elliptic space. Bibliography: 12 titles.
Keywords: Kahn's realisability problem of groups, group of homotopy self-equivalences, Anick's $R$-local homotopy theory. 
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M. Benkhalifa. Every group is the group of self-homotopy equivalences of finite dimensional $\mathrm{CW}$-complex. Sbornik. Mathematics, Tome 215 (2024) no. 9, pp. 1182-1201. http://geodesic.mathdoc.fr/item/SM_2024_215_9_a2/

[1] D. J. Anick, “Hopf algebras up to homotopy”, J. Amer. Math. Soc., 2:3 (1989), 417–453 | DOI | MR | Zbl

[2] D. J. Anick, “An $R$-local Milnor–Moore theorem”, Adv. Math., 77:1 (1989), 116–136 | DOI | MR | Zbl

[3] D. J. Anick, “$R$-local homotopy theory”, Homotopy theory and related topics (Kinosaki 1988), Lecture Notes in Math., 1418, Springer-Verlag, Berlin, 1990, 78–85 | DOI | MR | Zbl

[4] M. Benkhalifa, “Realisability of the group of self-homotopy equivalences and local homotopy theory”, Homology Homotopy Appl., 24:1 (2022), 205–215 | DOI | MR | Zbl

[5] M. Benkhalifa, “On the group of self-homotopy equivalences of an elliptic space”, Proc. Amer. Math. Soc., 148:6 (2020), 2695–2706 | DOI | MR | Zbl

[6] M. Benkhalifa and S. B. Smith, “The effect of cell-attachment on the group of self-equivalences of an $R$-localized space”, J. Homotopy Relat. Struct., 10:3 (2015), 549–564 | DOI | MR | Zbl

[7] P. J. Chocano, M. A. Morón and F. Ruiz del Portal, “Topological realizations of groups in Alexandroff spaces”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115:1 (2021), 25, 20 pp. | DOI | MR | Zbl

[8] C. Costoya, D. Méndez and A. Viruel, “Realisability problem in arrow categories”, Collect. Math., 71:3 (2020), 383–405 | DOI | MR | Zbl

[9] C. Costoya and A. Viruel, “Every finite group is the group of self-homotopy equivalences of an elliptic space”, Acta Math., 213:1 (2014), 49–62 | DOI | MR | Zbl

[10] J. de Groot, “Groups represented by homeomorphism groups. I”, Math. Ann., 138 (1959), 80–102 | DOI | MR | Zbl

[11] P. Hell and J. Nešetřil, Graphs and homomorphisms, Oxford Lecture Ser. Math. Appl., 28, Oxford Univ. Press, Oxford, 2004, xii+244 pp. | DOI | MR | Zbl

[12] D. W. Kahn, “Realization problems for the group of homotopy classes of self-equivalences”, Math. Ann., 220:1 (1976), 37–46 | DOI | MR | Zbl