Homotopy-commutativity in rotation groups (Q5894640)

This paper contributes to the question for which pairs \((m,n)\) of natural numbers greater than 1 the subgroups \(\text{SO}(m)\) and \(\text{SO}(n)\) of \(\text{SO}(m+n-1)\) commute (elementwise) up to homotopy, and it enlarges known results about this problem in a significant way. This question was raised by \textit{I. M. James} and \textit{E. Thomas} [Topology 1, 121-124 (1962; Zbl 0114.39204)], and they showed that this homotopy commutativity property does not hold for \(m+n\neq 4, 8\) if \(n\) or \(m\) is even or if the greatest power \(d(m)\) of 2 dividing \(q-1\) equals \(d(n)\). Note that for \(m+n\in \{4,8\}\) the mentioned property holds since then \(\text{SO}(m+n-1)\) is a retract of \(\text{SO}(m+n)\) and since the subgroups \(\text{SO}(m)\times\{\text{id}\}\) and \(\{\text{id}\}\times\text{SO}(n)\) of \(\text{SO}(n+m)\) centralize each other. The author sharpens the result by James and Thomas in the case that \(n\) and \(m\) are odd. He shows that the homotopy commutativity property in question does not hold for \(m\), \(n\) odd if \({m+n-2\choose n-1}\) is an even number. In fact, this sufficient condition for noncommutativity is proved to be necessary and sufficient for a weaker noncommutativity property. For this, the author considers the real projective space \(\mathbb{R}\mathbb{P}^{m-1}\) as a subset of \(\text{SO}(m)\), consisting of the products \(ii_0\), where \(i\) runs through all orthogonal reflections of \(\mathbb{R}^m\) at hyperplanes and \(i_0\) is a fixed hyperplane reflection. The main theorem of the paper then says that for \(m\), \(n\) odd the subsets \(\mathbb{R}\mathbb{P}^{m- 1}\subseteq\text{SO}(m)\) and \(\mathbb{R}\mathbb{P}^{n- 1}\subseteq\text{SO}(n)\) commute elementwise up to homotopy in \(\text{SO}(m+n -1)\) if and only if \({m+ n-2\choose n-1}\) is an odd number. For the proof, the fibration \(p:{\mathbf S}{\mathbf O}\to {\mathbf S}{\mathbf O}/\text{SO}(m+ n-1)\) with fibre \(\text{SO}(n+ m-1)\) and the associated sequence \[ \Omega{\mathbf S}{\mathbf O}@>\Omega p>>\Omega({\mathbf S}{\mathbf O}/\text{SO}(m+ n-1))@>\delta>>\text{SO}(m+ n-1)@>\iota>>{\mathbf S}{\mathbf O} \] is used. Since \(\text{SO}(m)\) and \(\text{SO}(n)\) commute in \(\text{SO}(m+ n)\subseteq{\mathbf S}{\mathbf O}\) up to homotopy, the commutator map \(c:\text{SO}(m)\wedge \text{SO}(n)\to \text{SO}(m+ n-1)\) has the property that \(\iota\circ c\) is null-homotopic, which means that \(c\) has a lift \[ \lambda_0: \text{SO}(m)\wedge \text{SO}(n)\to \Omega({\mathbf S}{\mathbf O}/\text{SO}(m+ n-1)). \] This lift was explicitly constructed by \textit{R. Bott} [Comment. Math. Helv. 34, 249-256 (1960; Zbl 0094.01503)]. The main theorem is about criteria for the restriction \(\lambda=\lambda_0|_{\mathbb{R}\mathbb{P}^{m-1}\wedge\mathbb{R}\mathbb{P}^{n-1}}\) to have a lift to \(\Omega{\mathbf S}{\mathbf O}\). By Whitehead's theorem, it can be established that this is equivalent to the existence of a map \(x:\mathbb{R}\mathbb{P}^{m-1}\wedge \mathbb{R}\mathbb{P}^{n-1}\to \Omega_0({\mathbf S}{\mathbf O})\) such that \(x^*(\alpha_{m+n-2})= \tau^{m-1}\otimes \tau^{n-1}\), where \(\alpha_{m+n-2}\) is the generator of degree \(m+n-2\) of the cohomology ring \(H^*(\Omega_0{\mathbf S}{\mathbf O})= \mathbb{Z}/2\mathbb{Z}[\alpha_2,\alpha_4,\dots]/(\alpha_{4k}- \alpha^2_{2k})\) and \(\tau\) is the generator of \(H^1(\mathbb{R}\mathbb{P}^{m-1}, \mathbb{Z}/2\mathbb{Z})\) and of \(H^1(\mathbb{R}\mathbb{P}^{n- 1},\mathbb{Z}/2\mathbb{Z})\), respectively. The main result is that such a map exists if and only if \({m+ n-2\choose n-1}\) is odd. In order to show that such a map cannot exist for \({m+ n-2\choose n-1}\) even (with \(m\), \(n\) odd), the author proves results under this assumption about the cohomology classes \(x^*(\alpha_{2k})\) for arbitrary maps \(x: \mathbb{R}\mathbb{P}^{m-1}\wedge \mathbb{R}\mathbb{P}^{n-1}\to \Omega_0({\mathbf S}{\mathbf O})\), using Steenrod squaring operations. In particular, he obtains that then \(x^*(\alpha_{m+ n-2})= 0\) for every such map.