Topology of spaces of functions without compound singularities
The author considers spaces of smooth mappings not having too complicated singularities, and he studies the cohomology of such spaces. Let \({\mathfrak A}\) be a closed, semialgebraic singularity subset of the jetspace \(J^ k({\mathbb{R}}^ m,{\mathbb{R}}^ n)\) and consider a smooth map \(\phi: M\to {\mathbb{R}}^ n\) from an m-dimensional manifold not having singularities of type \({\mathfrak A}\) in a neighbourhood of \(\partial M\). Let \(A=A(M,{\mathfrak A},\phi)\) be the space of maps coinciding with \(\phi\) in a neighbourhood of \(\partial M\) and not having singularities of type \({\mathfrak A}\) at any point, and let \(B=B(M,{\mathfrak A},\phi)\) be the space of sections in \(J^ k(M,{\mathbb{R}}^ n)\) coinciding with \(j^ k\phi\) in a neighbourhood of \(\partial M\), and avoiding the corresponding singularity subset \({\mathfrak A}(M)\). Then it is proven that if \(co\dim {\mathfrak A}\geq m+2,\) the imbedding \(j^ k: A\to B\) given by k-jet extension induces an isomorphism of the cohomology rings of A and B, and under certain additional conditions on \({\mathfrak A}\), m and n, the above map is also a weak homotopy-equivalence. It is also proven that the space of one- variable functions identically equal 1 outside some compact and without k-fold zeros, is weakly homotopically equivalent to the loop space \(\Omega S^{k-1}\). In addition, the author calculates the multiplicative structure in the cohomology ring of the space of one variable polynomials of degree d not having roots of multiplicity k.
scientific article; zbMATH DE number 4209117