Bordism theory and homotopy properties of supermanifolds
The definition of a smooth supermanifold with boundary is given. It is analogous to the definition of a smooth supermanifold given in: \textit{F. A. Berezin}, Introduction to the algebra and analysis with anticommuting variables (Russian) (MGU 1983; Zbl 0527.15020). The latter is modeled on the pair \((R^ p,U^{p,q})\), where \(U^{p,q}\) is a sheaf defined as follows: for any open set, \(U\subset R^ p\) \(U^{p,q}(U)\) is the real algebra of all smooth functions on U with the values in the fixed exterior algebra \(\Lambda_ p\); to define the supermanifold with boundary, \(R^ p_+\) is used instead of \(R^ p.\) The boundary of the supermanifold with boundary is defined. The boundary is again a smooth supermanifold. The subtle point is that in general, given any atlas, the topological boundary of the underlying space does not define the boundary of the supermanifold which is also a smooth supermanifold. However, this is the case for the so called simple atlas, i.e. the one for which the transition functions take the special form: the even and odd variables transform independently and, moreover, the odd ones transform linearly. The definition of the boundary given by the authors is very natural and has an important advantage that all boundaries of a given supermanifold with boundary are isomorphic to supermanifolds. Several results concerning the homotopy and bordism theories of supermanifolds are announced. They seem to be rather straightforward generalizations of the ideas used in ``even'' mathematics.
scientific article; zbMATH DE number 4067899
scientific article