For the Stiefel manifold \(V_k(\mathbb C^n)\) of orthonormal \(k\)-frames in \(\mathbb C^n\) and the Grassmann manifold \(G_k(\mathbb C^n)\) of \(k\)-dimensional subspaces of \(\mathbb C^n\) the author considers the natural \(U(k)\)-bundle \(p:V_k(\mathbb C^n)\rightarrow G_k(\mathbb C^n)\); in particular, its rational relative evaluation subgroup.
For a space \(X\) and a positive integer \(n\), the evaluation subgroup \(G_n(X)\) of the homotopy group \(\pi_n(X)\) was introduced by D. H. Gottlieb in [Am. J. Math. 91, 729–756 (1969; Zbl 0185.27102)]. A generalization of this concept appeared in [M. H. Woo and J.-R. Kim, J. Korean Math. Soc. 21, 109–120 (1984; Zbl 0559.55017)], where, for a map \(f:X\rightarrow Y\), the authors defined a subgroup \(G_n(Y,X;f)\) of \(\pi_n(Y)\) (in the case \(X=Y\) and \(f=1_X\), \(G_n(X,X;1_X)=G_n(X)\)). There are also the so-called relative evaluation subgroups (or relative Gottlieb subgroups) \(G_n^{\mathrm{rel}}(Y,X;f)\), which in some cases fit into a long exact sequence of the form \[ \cdots\rightarrow G_{n+1}^{\mathrm{rel}}(Y,X;f)\rightarrow G_n(X)\rightarrow G_n(Y,X;f)\rightarrow G_n^{\mathrm{rel}}(Y,X;f)\rightarrow\cdots. \] The rational relative evaluation subgroup of a map \(f:X\rightarrow Y\) is the group \(G_n^{\mathrm{rel}}(Y_{\mathbb Q},X;h\circ f)\), where \(h:Y\rightarrow Y_{\mathbb Q}\) is the rationalization.
In the paper under review the author uses Sullivan minimal models for simply connected spaces and maps between them, as well as the notion of the derivation of a map between commutative differential graded algebras, to obtain some results concerning the rational relative evaluation subgroup of the bundle \(p:V_k(\mathbb C^n)\rightarrow G_k(\mathbb C^n)\). It is stated in the abstract that the isomorphism \(G_*(G_k(\mathbb C^n),V_k(\mathbb C^n);p)\cong G_*^{\mathrm{rel}}(G_k(\mathbb C^n),V_k(\mathbb C^n);p)\oplus G_*(V_k(\mathbb C^n))\) is proven in the paper (although the reviewer was not able to find this in the body of the paper).