Equivariant cobordism of Grassmann and flag manifolds
Let \(G\) be a compact Lie group. A smooth \(G\)-manifold \(M\) with a given action of \(G\) is denoted by \((M,\Phi)\), where \(\Phi:G\times M\to M\) desnotes the action map. An element \(x\in M\) is called a stationary point if \(\Phi(g,x)=x\) for all \(g\in G\). A smooth closed \(n\)-dimensional \(G\)-manifold \((M^n,\Phi)\) with finite stationary point set, is said to bound equivariantly if and only if there is an action \((W^{n+1},\Phi)\) on a compact \((n+1)\)-manifold for which the induced action \((\partial W^{n+1},\Phi/\partial W^{n+1})\) is equivariantly diffeomorphic to \((M^n,\Phi)\). Let \(O(n)\) denote the orthogonal group of \(n\times n\) matrices. The subgroup of \(O(n)\) consisting of diagonal matrices can be identified with \((\mathbb{Z}_2)^n\). Let \(e_1,\dots, e_n\) be the standard basis of \(\mathbb{R}^n\), and \(T_j\) the involution \[ T_j(e_i)= \begin{cases} -e_i &\text{if } i=j,\\ e_i &\text{if }i\neq j.\end{cases} \] Then there exists an action of \((\mathbb{Z}_2)^n\) on \(\mathbb{R}^n\) given by pairwise commuting actions of \(T_i\)'s. This action induces an action of \((\mathbb{Z}_2)^n\) on \(G_{n,k}\), the Grassmann manifold of \(k\)-dimensional subspaces in \(\mathbb{R}^n\), and this action has finite stationary point set. A \(k\)-plane \(X\) in \(\mathbb{R}^n\) is fixed by this action if and only if \(X=\langle e_{i_1},\dots, e_{i_k}\rangle=: E_\alpha\), \(\alpha:= 1\leq i_1<i_2<\dots< i_k\leq n\). Thus there are \({n\choose k}\) stationary points for this action. A Grassmann manifold \(G_{n,k}\) along with this action of \((\mathbb{Z}_2)^n\) will be denoted by \((G_{n,k},\Phi)\). The author proves: Theorem 1.1. (a) \((G_{n,k},\Phi)\) bounds equivariantly if \(n=2k\). (b) \((G_{n,k},\Phi)\) does not bound equivariantly if \(n\neq 2k\). Then, let \(G(n_1, n_2,\dots, n_s)\), \(n=n_1+ \cdots+n_s\), \(s\geq 3\) denote the real flag manifold of all flags \((A_1, A_2,\dots, A_s)\) where \(A_i\) is a left vector subspace of \(\mathbb{R}^n\), \(A_i\perp A_j\) for \(i\neq j\), \(\dim_R A_i=n_i\), \(1\leq i,j\leq s\), \(G(n_1, n_2,\dots, n_s)\) is a smooth manifold of dimension \(\sum_{1\leq i<j\leq s}n_in_j\). Alternatively, it can be described as the homogeneous space \(O(n)/ O(n_1)\times \dots\times O(n_1)\). The group \((\mathbb{Z}_2)^n\) acts on \(G(n_1,\dots, n_s)\) by pairwise commutating involutions \(T_\alpha\), \(\alpha=1,\dots, n\), having finite stationary point set. This action is induced from the action of the \(T_\alpha\)'s on \(\mathbb{R}^n\) as described in the last section. This action is denoted by \((G(n_1, n_2,\dots, n_s),\Phi)\). Then the author proves: Theorem 1.2. \((G(n_1, n_2,\dots n_s),\Phi)\) bounds equivariantly if and only if \(n_i=n_j\) for some \(i\), \(j\), \(i\neq j\). These results have their origin in work of \textit{P. E. Conner} [Differentiable periodic maps. Lect. Notes Math. 738 (1979; Zbl 0417.57019)], \textit{P. Sankaran} [Can. Math. Bull. 34, 119-122 (1991; Zbl 0667.57020)] and in a review by \textit{R. E. Stong} [MR 89d:57050].
scientific article; zbMATH DE number 903373
scientific article