A survey of Anick-Gray-Theriault constructions and applications to the exponent theory of spheres and Moore spaces (Q2707512)

All spaces are assumed to be pointed, connected, \(p\)-local, and of the homotopy type of a finite CW-complex. In [Differential algebras in topology, Res. Notes Math. 3 (1993; Zbl 0770.55001)] \textit{D. Anick} constructed a homotopy fibration sequence of \(H\)-spaces NEWLINE\[NEWLINE\Omega^2 S^{2n+1} @>\Phi_r>> S^{2n-1}\to T^{2n-1}(p^r) \to\Omega S^{2n+1}NEWLINE\]NEWLINE for \(p\geq 5\), \(r\geq 1\), \(n\geq 1\), such that \(\Phi_r \circ E^2\simeq p^r\) and \(E^2\circ \Phi_r\simeq \Omega^2p^r\), where \(E^2:S^{2n+1} \to\Omega^2S^{2n+1}\) is the double suspension. In [Topology 34, No. 4, 859-881 (1995; Zbl 0868.55006)] \textit{D. Anick} and \textit{B. Gray} showed that the maps involved were \(H\)-maps. In a series of papers [Properties of Anick's spaces, Trans. Am. Math. Soc. 353, No. 3, 1009-1037 (2000; Zbl 0992.55008), and ibid., No. 4, 1551-1566 (2001; Zbl 0961.55013)] \textit{St. D. Theriault} extended this construction to the case \(p\geq 3\), showed that \(T^{2n-1}(p^r)\) is homotopy associative and commutative for \(p\geq 5\) and that it satisfies a universal property for \(p\geq 3\). This led the author to the following definition: An Anick-Gray-Theriault construction is a \(p\)-local homotopy associative and commutative \(H\)-space together with a map \(i:X\to A(X)\) satisfyingNEWLINENEWLINENEWLINE(i) any map \(f:X\to G\) into a homotopy associative and commutative \(H\)-space \(G\) extends to an \(H\)-map \(\overline f:A(X)\to G\), uniquely up to homotopy;NEWLINENEWLINENEWLINE(ii) \(H_*(A(X); \mathbb{F}_p)\) is the graded symmetric algebra generated by \(\widetilde H_*(X; \mathbb{F}_p)\), the reduced homology of \(X\).NEWLINENEWLINENEWLINEThe paper is a survey of known results about the existence and non-existence of Anick-Gray-Theriault-constructions. These are used to derive information about homotopy groups of spheres and Moore spaces, in particular their exponents.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].