Lie-model for Thom spaces of tangent bundles (Q2790288)

Let \(M\) be a closed, connected, oriented manifold, \(F(M, 2)\) the configuration space of two points in \(M\) and \(\text{Th}(TM)\) the Thom space of the tangent bundle over \(M\).NEWLINENEWLINEIn Section 1 of the paper under review, observing a homotopy cofibre sequence NEWLINE\[NEWLINEF(M, 2) \to M\times M \to \text{Th}(TM)NEWLINE\]NEWLINE with the projection \(\rho : M\times M \to \text{Th}(TM)\), the authors investigate the map \(\rho_*\) induced in homology. In fact, the composite \(Th_*\circ \rho_* : H_*(M\times M) \to H_*(D(\nu_\Delta))\) with the Thom isomorphism is described in terms of intersection products, where \(D(\nu_\Delta)\) is the disk bundle of the normal bundle of the diagonal embedding \(\Delta : M \to M\times M\). Section 2 gives a rational commutative model of the Thom space \(\text{Th}(\xi)\) of an oriented vector bundle \(\xi\) over \(M\) provided \(M\) is nilpotent. A model for the (homotopy) cofibre of a map \(f : X \to Y\) is induced by the kernel of the so-called \textit{surjective model} for \(f\). By applying the key fact in rational homotopy theory to an appropriate surjective model for the sphere bundle associated to \(\xi\), the commutative model for \(\text{Th}(\xi)\) is obtained. Section 3 is devoted to constructing the minimal Lie model of \(\rho\) by using the commutative model for the Thom space discussed in Section 2. In consequence, we see that the Lie model is determined explicitly by the intersection product on \(H_*(M)\) if \(M\) is simply-connected. The authors conjecture that a model of the configuration space \(F(M, 2)\) is constructed by using the kernel of the Lie model of \(\rho\) in indecomposable elements with an explicit form. Section 4 considers examples of the minimal Lie model of \(\rho\) in case of spheres. The paper concludes with an example which says that the conjecture mentioned above holds for the \(2\)-dimensional complex projective space.