Constructions of self-maps of \(\operatorname{SU}(4)\) via Postnikov towers (Q2183503)

Let \(M\) be a closed oriented \(n\)-dimensional manifold, and let \([M]\in H_n(M)\) be the fundamental class of \(M\). Then the degree \(\text{deg}(f)\) of a self map \(f:M\to M\) is the integer \(t\) such that \(f_*([M])=t[M]\). Let \(D(M)=\{\text{deg}(f)\mid f:M\to M\}\) be the set of degrees of self maps of \(M\). In this paper, the authors study \(D(\operatorname{SU}(4))\). Let \(f\) be a self map of \(\operatorname{SU}(4)\), and let \(x_3,x_5\) and \(x_7\) be generators of \(H^*(\operatorname{SU}(4))\cong\Lambda[x_3,x_5,x_7]\) with degrees \(3,5\) and \(7\). Then \(f^*(x_i)=t_ix_i\) for some number \(t_i\) and \(i\in\{3,5,7\}\). The tuple \((t_3,t_5,t_7)\) is called the multidegree of \(f\). In Theorem \(A\), the authors show that there is a self map \(f:\operatorname{SU}(4)\to \operatorname{SU}(4)\) with multidegree \((t_3,t_5,t_7)\) if and only if \[ \begin{array}{c c c r} t_3\equiv t_5\pmod{2} &\text{and} &t_3\equiv t_7\pmod{6}. &(1) \end{array} \] Theorem \(A\), together with the fact that \(\text{deg}(f)=t_3t_5t_7\), implies Corollary \(B\), which says that \(D(\operatorname{SU}(4))\) consists of odd numbers and multiples of 8 only. In Section 2, the authors show the sufficiency of Theorem \(A\). In Sections 3 and 4, they recall the key facts of the Whitehead sequence, Postnikov systems and cohomology operations, and prove the necessity of Theorem \(A\) by constructing a self map with multidegree \((t_3,t_5,t_7)\) for any tuple \((t_3,t_5,t_7)\) satisfying (1). In Section 5 they give an alternative proof to Corollary \(B\) directly without looking into multidegrees of self maps. They construct a self map with degree \(t\) where \(t\) is an odd number or a multiple of 8. In Section 6, the authors mention how to modify their construction of self maps \(f:\operatorname{SU}(4)\to \operatorname{SU}(4)\) to \(f\) an \(H\)-map. Also, as their method can be applied to other Lie groups, they conjecture that for \(n\geq5\), the set of multidegrees of self maps \(\operatorname{SU}(n)\to \operatorname{SU}(n)\) consists of tuples \((t_3,\ldots,t_{2n-1})\) satisfying congruence conditions \(t_{ik}\equiv t_{j_k}\pmod{m_k}\) for some indices \(i_k\) and \(j_k\) and moduli \(m_k\in\mathbb{Z}\).