The effect of cell-attachment on the group of self-equivalences of an elliptic space. (English) Zbl 1497.55015
A simply connected rational space \(X\) is called elliptic if the two graded vector spaces \(H^*(X;\mathbb{Q})\) and \(\pi_*(X)\) are finite dimensional, and the number \(m=\max\{i:H^i(X;\mathbb{Q})\not= 0\}\) is called the formal dimension of \(X\). Let \(\mathcal{E}(X)\) denote the group of homotopy classes of self-homotopy equivalences of \(X\), and \(\mathcal{E}_*(X)\) denotes the subgroup of \(\mathcal{E}(X)\) consisting of all \(f\in \mathcal{E}(X)\) inducing the identity on homology groups.
From now on, let \(X\) be a simply connected rational elliptic space of formal dimension \(m\), and let \(Y\) be the space obtained by attaching rational cells of dimension \(q\) to \(X\). In this paper the author studies the relation between \(\mathcal{E}(X)\) and \(\mathcal{E}(Y)\) (resp. \(\mathcal{E}_*(X)\) and \(\mathcal{E}_*(Y)\)) by using the Quillen model of rational homotopy theory. In particular, he proves that there are isomorphisms \(\mathcal{E}(Y)\cong \mathrm{GL}(r,\mathbb{Q})\times\mathcal{E}(X)\) and \(\mathcal{E}_*(Y)\cong\mathcal{E}_*(X)\) if \(q>2m+1\) and \(r=\dim H_q(X;\mathbb{Q})\). As an application, he also proves that for any finite group \(G\) and any positive integer \(r\), there exists a simply connected space \(X\) such that \(\mathcal{E}(X)\cong \mathrm{GL}(r,\mathbb{Q})\times G\). His proof is based on the analysis of the Quillen model of \(\mathcal{E}_*(X)\).
From now on, let \(X\) be a simply connected rational elliptic space of formal dimension \(m\), and let \(Y\) be the space obtained by attaching rational cells of dimension \(q\) to \(X\). In this paper the author studies the relation between \(\mathcal{E}(X)\) and \(\mathcal{E}(Y)\) (resp. \(\mathcal{E}_*(X)\) and \(\mathcal{E}_*(Y)\)) by using the Quillen model of rational homotopy theory. In particular, he proves that there are isomorphisms \(\mathcal{E}(Y)\cong \mathrm{GL}(r,\mathbb{Q})\times\mathcal{E}(X)\) and \(\mathcal{E}_*(Y)\cong\mathcal{E}_*(X)\) if \(q>2m+1\) and \(r=\dim H_q(X;\mathbb{Q})\). As an application, he also proves that for any finite group \(G\) and any positive integer \(r\), there exists a simply connected space \(X\) such that \(\mathcal{E}(X)\cong \mathrm{GL}(r,\mathbb{Q})\times G\). His proof is based on the analysis of the Quillen model of \(\mathcal{E}_*(X)\).
Reviewer: Kohhei Yamaguchi (Tokyo)
MSC:
| 55P10 | Homotopy equivalences in algebraic topology |
| 55P62 | Rational homotopy theory |
| 55P15 | Classification of homotopy type |
References:
| [1] | M. Arkowitz and G. Lupton, Rational obstruction theory and rational homotopy sets, Math. Z. 235 (2000), no. 3, 525-539. · Zbl 0968.55005 |
| [2] | M. Benkhalifa, Rational self-homotopy equivalences and Whitehead exact sequence, J. Homotopy Relat. Struct. 4 (2009), no. 1, 135-144. · Zbl 1190.55006 |
| [3] | M. Benkhalifa, Realizability of the group of rational self-homotopy equivalences, J. Homotopy Relat. Struct. 5 (2010), no. 1, 361-372. · Zbl 1278.55021 |
| [4] | M. Benkhalifa, On the group of self-homotopy equivalences of an elliptic space, Proc. Amer. Math. Soc. 148 (2020), 2695-2706. · Zbl 1437.55012 |
| [5] | M. Benkhalifa and S. B. Smith, The effect of cell attachment on the group of self-equivalences of an R-local space, J. Homotopy Relat. Struct. 10 (2015), no. 3, 549-564. · Zbl 1326.55008 |
| [6] | C. Costoya and A. Viruel, Every finite group is the group of self homotopy equivalences of an elliptic space, Acta Math. 213 (2014), 49-62. · Zbl 1308.55005 |
| [7] | Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Grad. Texts in Math., 205, Springer, New York, 2001. · Zbl 0961.55002 |
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