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The ring of stable homotopy classes of self-maps of \(A_n^2\)-polyhedra. (English) Zbl 1458.55005

For a connected space \(X\), let \([X,X]\) denote the monoid consisting of all homotopy classes of self-maps over \(X\) whose multiplication is induced from the composition of maps, and let \(\displaystyle \{X,X\}=\lim_{n\to\infty}[\Sigma^nX,\Sigma^nX]\) denote the ring of stable maps whose addition is induced from the track addition. Recall that a space \(X\) is called an \(A^2_n\)-polyhedron if it is an \((n+2)\)-dimensional \((n-1)\)-connected CW complex. From now on, let \(n\geq 3\) and let us consider an \(A^2_n\)-polyhedron \(X\) together with the Whitehead exact sequence \[ H_{n+2}(X;\mathbb{Z}) \stackrel{b_{n+2}}{\longrightarrow} H_n(X;\mathbb{Z})\otimes\mathbb{Z}/2 \stackrel{i_n}{\longrightarrow} \pi_{n+1}(X) \stackrel{h_{n+1}}{\longrightarrow} H_{n+1}(X;\mathbb{Z}) \longrightarrow 0 \tag{1} \] Since \(n\geq 3\), the space \(X\) is in the stable range and we write \(\mathrm{End} (X)=\{X,X\}\). Baues proved that the homotopy types of \(A^2_n\)-polyhedra \(X\) are classified by isomorphism classes of triples of abelian groups, \[ (H_{n+2}(X;\mathbb{Z}),H_{n+1}(X;\mathbb{Z}),H_{n}(X;\mathbb{Z})) \] together with the exact sequence \((1)\). As its generalization, consider the category \(\Gamma\text{-sequences}^{n+2}\) whose objects are triples \((H_{n+2},H_{n+1},H_n)\), of abelian groups with \(H_{n+2}\) free, together with an exact sequence \[ H_{n+2} \longrightarrow H_n\otimes\mathbb{Z}/2 \longrightarrow \pi_{n+1} \longrightarrow H_{n+1} \longrightarrow 0. \] If \(\mathcal{A}^2_n\) denotes the homotopy category of \(A^2_n\)-polyhedron \(X\), we naturally obtain the functor \(\Gamma :\mathcal{A}^2_n\to \Gamma\text{-sequences}^{n+2}\) given by \[ \Gamma (X)=(H_{n+2}(X;\mathbb{Z}),H_{n+1}(X;\mathbb{Z}),H_{n}(X;\mathbb{Z})) \] together with (1). Moreover, we also obtain a natural ring homomorphism \[ H:\mathrm{End}(X)\to \mathrm{End}(\Gamma (X));\quad \alpha\mapsto (H_{n+2}(\alpha),H_{n+1}(\alpha),H_n(\alpha)). \]
In this paper, the author proves that the homomorphism \(H:\mathrm{End}(X)\to \mathrm{End}(\Gamma (X))\) is an epimorphism whose kernel consists in the homologically trivial self-maps of \(X\). Moreover, as an application he also studies the realizability problem for \(\mathrm{End}(\Gamma (X))\). In particular, he proves that there is an \(A^2_n\)-polyhedron \(X\) in the stable range such that \(\mathrm{End}(\Gamma (X))\cong R\) if \(R\) is a ring such that \(R\cong \mathrm{End}(G_1)\times \mathrm{End}(G_2)\times \mathrm{End}(G_3)\) for abelian groups \(G_1\), \(G_2\) and a free abelian group \(G_3\). When \(n\geq 3\) and \(p\geq 2\) is a prime, he also shows that there is no \(A^2_n\)-polyhedron \(X\) of finite type such that \(\mathrm{End}(\Gamma (X))\cong \mathbb{F}^3_p.\)

MSC:

55P10 Homotopy equivalences in algebraic topology
55P40 Suspensions
55Q52 Homotopy groups of special spaces
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References:

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