\(K\)-theory for conical isolated singularities. (\(K\)-théorie pour les singularités coniques isolées). (Q2577022)

Let \(X\) be a compact and path-connected topological space, and let \( A=\left\{ a_{i}\in X| i\in \mathcal{I}\right\} \) be a collection of isolated points in such a way that \(X-A\) is an open \(C^{\infty }\)-manifold. We say that \(X\) is a manifold with isolated conical singularities if each \(a_{i}\in A\) has a neighbourhood \(V_{i}\) such that for a given compact smooth manifold \( L_{i}\) there is a homeomorphism \(\varphi _{i}:V_{i}\rightarrow \left( \left[ 0,\varepsilon _{i}\right) \times L_{i}\right) /\left( \left\{ 0\right\} \times L_{i}\right) \) that induces a diffeomorphism \(\left( V_{i}-\left\{ a_{i}\right\} \right) \rightarrow \left( 0,\varepsilon _{i}\right) \times L_{i}\). We write \(\widetilde{X}\) for the manifold obtained from \(X\) by deleting the conical neighbourhoods and glueing the cylinders \(\left[ 0,\varepsilon _{i}\right) \times L_{i}\) instead. Thus \(\widetilde{X}\) is a manifold with boundary and \(\partial \widetilde{X}=\cup _{i}L_{i}\). Let \(q<\dim X\) be a positive integer; an admissible triplet is a triplet \( \left( E,F,t\right) \) such that: \(\left( i\right) \) \(E\rightarrow \widetilde{ X}\) is a complex bundle, \(q\)-flat on \(\partial \widetilde{X}\); \(\left( ii\right) \) \(F\rightarrow \widetilde{X}\) is a sub-bundle with ch\(_{j}F=\)ch\(_{j}E\) for \( 2j\geq q+2\) and ch\(_{j}F=0\) for \(0<2j\leq q+1\); \(\left( iii\right) \) \( F_{| \partial \widetilde{X}}\) is trivial and \(t\) is a trivialisation. In this paper the authors define \(K\)-theoretical groups \(I_{q}K\left( X\right) \) as the Grothendieck construction of the additive semigroup [\textit{M. F. Atiyah} and \textit{D. W. Anderson}, \(K\)-theory. With reprints of \textit{M. F. Atiyah}: Power operations in K-theory. New York-Amsterdam: W. A. Benjamin, Inc. (1967; Zbl 0159.53302)] generated by the isomorphism classes \(\left[ E,F,t\right] \). By using the properties of intersection cohomology [\textit{A. Borel}, et. al., Intersection cohomology. (Notes of a Seminar on Intersection Homology at the University of Bern, Switzerland, Spring 1983). Progress in Mathematics, Vol. 50. Swiss Seminars. (Boston-Basel-Stuttgart): Birkhäuser. (1984; Zbl 0553.14002)], the Chern character ch\(^{\prime }:I_{q}K\left( X\right) \rightarrow IH_{\widetilde{p}}^{\text{even}}\left( X; {\mathbb Q}\right) \) is defined obtaining an isomorphism \[ ch^{\prime }:I_{q}K\left( X\right) \otimes {\mathbb Q}\rightarrow IH_{ \widetilde{p}}^{\text{even}}\left( X;{\mathbb Q}\right) \] which is the main result in this note, where \(\widetilde{p}\) is a perversity and \(\widetilde{q}\) the complementary perversity. Finally, using the multiplicative \(K\)-theoretical groups of Karoubi [\textit{M. Karoubi}, Homologie cyclique et K-theorie, Astérisque, 149. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. (1987; Zbl 0648.18008)], the authors construct the Chern-Weil version of the above construction.