The number of summands in \(v_1\)-periodic homotopy groups of \(SU(n)\) (Q817630)

This paper is based on the author's thesis [``The number of summands in \(v_1\)-periodic homotopy groups of \(SU(n)\)'', Ph.D. Thesis, Lehigh University, 2004]. The author determines the number of summands of the \(v_1\)-periodic homotopy group \(v_1^{-1}\pi_{2k-1}(SU(n); p)\) for all \(p, k\) and \(n\), where \(p\) is an odd prime. In the present case this number coincides with the \(p\)-rank of \(v_1^{-1}\pi_{2k-1}(SU(n); p)\), (abbreviated as \(\text{rank}_p(v_1^{-1}\pi_{2k-1}(SU(n); p))\)). The result is \[ \text{rank}_p(v_1^{-1}\pi_{2k-1}(SU(n); p))=c(p, k, n) \] where \(c(p, k, n)=\lfloor\log_p(n/(\bar{k}+1))\rfloor + \varepsilon(p, k, n)\), and \(\varepsilon(p, k, n)=0\) or \(1\) according as it holds that \(\bar{k}=1\), \(p^e+p^{e-1}-1 \leq n \leq p^e+1\) or does not, for some integer \(e\geq 2\) with \(\bar{k}\equiv k \bmod p-1\), \(1 \leq \bar{k} \leq p-1\). This work is motivated by the result of \textit{D. M. Davis} [J. Lond. Math. Soc., II. Ser. 43, 529--544 (1991; Zbl 0760.55013)] stating that if \(p\) is an odd prime, then \(v_1^{-1}\pi_{2k}( \text{SU}(n); p)\) is a cyclic group of order \(p^{e_p(k, n)}\) for a certain integer \(e_p(k, n)\) (the explicit formula for which is omitted as it is too complicated to be described here), and \(v_1^{-1}\pi_{2k-1}(\text{SU}(n); p)\) has the same order as \(v_1^{-1}\pi_{2k}(\text{SU}(n); p)\) but is not necessarily cyclic. From this result one knows that the group structure of \(v_1^{-1}\pi_{2k-1}( \text{SU}(n); p)\) still remains to be determined. The principal tool for the proof of the result is a characterization of \(v_1^{-1}\pi_{2k-1}(X; p)\) by \textit{A. K. Bousfield} [Topology 38, 1239--1264 (1999; Zbl 0933.57034)]. By applying this to the case \(X=\text{SU}(n)\) and using another algebraic result due to \textit{D. M. Davis} [Homology Homotopy Appl. 5, 297--324 (2003; Zbl 1031.55008)] it follows that \(v_1^{-1}\pi_{2k-1}(\text{SU}(n); p)\) is isomorphic to the group presented by a certain matrix \(M\) generated by the Adams operations. Also it can be observed that this fact yields \[ \text{rank}_p(v_1^{-1}\pi_{2k-1}(\text{SU}(n); p))=\text{corank}(M) \] where \(\text{corank}(M)\) denotes the number of columns in \(M\) minus the rank of \(M\). This shows that for the proof of the result it suffices to prove that \[ \text{corank}(M)=c(p, k, n). \] Most of the paper is devoted to establishing this equality. Actually this requires a fair amount of matrix-theoretical calculation.