On modular cohomotopy groups. (English) Zbl 1520.55011
The cohomotopy groups \(\pi^n (X) = [X, S^n]\) consisting of homotopy classes of base point preserving continuous maps from a CW-complex \(X\) into the \(n\)-sphere with \(\dim(X) \leq 2n-2\) were investigated by K. Borsuk [C. R. Acad. Sci., Paris 202, 1400–1403 (1936; Zbl 0014.03903)] and E. Spanier [Ann. Math. (2) 50, 203–245 (1949; Zbl 0032.12402)]. The generalized stable cohomotopy groups \(\pi^n (X,A;G) = [X,A ; M(G,n),*]\) of a CW-pair \((X,A)\) of dimension at most \(2n-2\) were developed by F. P. Peterson [Am. J. Math. 78, 259–281 (1956; Zbl 0071.38505)], where \(M(G,n)\) is the Moore space of type \((G,n)\).
In this paper under review, the authors focus on the study of the so-called modular cohomotopy groups \(\pi^n (X; \mathbb{Z}/p^r ) = [X, M(\mathbb{Z}/p^r, n)]\) consisting of homotopy classes of base point preserving continuous maps from a CW-complex \(X\) to the mod \(p^r\) Moore space \(M(\mathbb{Z}/p^r, n)\), where \(p\) is a prime number. The authors show that if \(X\) is a CW-complex of dimension at most \(n+2p-3\) and \(n \geq 2p-1\), then, for \(G = \mathbb{Z}_{(p)}\) or \(\mathbb{Z}/p^r\) with \(r \geq 1\), there is a short exact sequence \[ 0 \longrightarrow T_{\Omega \mathcal{P}^1_G}(X) \longrightarrow \pi^n(X,G) \stackrel{h^n_G}{\longrightarrow} H^n(X,G) \longrightarrow 0 \] of abelian groups, where \(\mathcal{P}^1_G : K(G,n) \longrightarrow K(\mathbb{Z}/p,n ) \stackrel{\mathcal{P}^1}{\longrightarrow} K(\mathbb{Z}/p, n+2p-2)\), and \(T_{\Omega \mathcal{P}^1_G}(X) = H^{n+2p-3} (X; \mathbb{Z}/p)/\mathcal{P}^1_G (H^{n-1}(X;G))\). They construct various short exact sequences with respect to cohomotopy groups, and give conditions for the splitness of the short exact sequences. The authors concretely prove that if \(X\) is a finite complex of dimension at most \(4n-3\) with \(n \geq 2\), then, for each prime \(p \geq 5\), there is an exact sequence \[ 0 \longrightarrow \pi^{2n-1} (X; \mathbb{Z}_{(p)}) \otimes \mathbb{Z}/p^r \longrightarrow \pi^{2n-1} (X; \mathbb{Z}/p^r) \longrightarrow \pi^{2n+1} (\Sigma X; \mathbb{Z}_{(p)}) \] of abelian groups as a generalization of Peterson’s exact sequence theorem for cohomotopy with \(\mathbb{Z}/p^r\)-coefficients.
In this paper under review, the authors focus on the study of the so-called modular cohomotopy groups \(\pi^n (X; \mathbb{Z}/p^r ) = [X, M(\mathbb{Z}/p^r, n)]\) consisting of homotopy classes of base point preserving continuous maps from a CW-complex \(X\) to the mod \(p^r\) Moore space \(M(\mathbb{Z}/p^r, n)\), where \(p\) is a prime number. The authors show that if \(X\) is a CW-complex of dimension at most \(n+2p-3\) and \(n \geq 2p-1\), then, for \(G = \mathbb{Z}_{(p)}\) or \(\mathbb{Z}/p^r\) with \(r \geq 1\), there is a short exact sequence \[ 0 \longrightarrow T_{\Omega \mathcal{P}^1_G}(X) \longrightarrow \pi^n(X,G) \stackrel{h^n_G}{\longrightarrow} H^n(X,G) \longrightarrow 0 \] of abelian groups, where \(\mathcal{P}^1_G : K(G,n) \longrightarrow K(\mathbb{Z}/p,n ) \stackrel{\mathcal{P}^1}{\longrightarrow} K(\mathbb{Z}/p, n+2p-2)\), and \(T_{\Omega \mathcal{P}^1_G}(X) = H^{n+2p-3} (X; \mathbb{Z}/p)/\mathcal{P}^1_G (H^{n-1}(X;G))\). They construct various short exact sequences with respect to cohomotopy groups, and give conditions for the splitness of the short exact sequences. The authors concretely prove that if \(X\) is a finite complex of dimension at most \(4n-3\) with \(n \geq 2\), then, for each prime \(p \geq 5\), there is an exact sequence \[ 0 \longrightarrow \pi^{2n-1} (X; \mathbb{Z}_{(p)}) \otimes \mathbb{Z}/p^r \longrightarrow \pi^{2n-1} (X; \mathbb{Z}/p^r) \longrightarrow \pi^{2n+1} (\Sigma X; \mathbb{Z}_{(p)}) \] of abelian groups as a generalization of Peterson’s exact sequence theorem for cohomotopy with \(\mathbb{Z}/p^r\)-coefficients.
Reviewer: Dae-Woong Lee (Jeonju)
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