turkmath.org

In this talk, we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. As an application, we deduce the existence of certain special 3 and 4-dimensional submanifolds of G_2 holonomy Riemannian manifolds with special properties. These are called Harvey-Lawson(HL) pairs. Which appeared first in the work of Akbulut & Salur about G_2 dualities. Another application is to the free embeddings. We show that if there is a coassociative-free embedding of a 4-manifold into the Euclidean 7-space then the signature vanishes along with the Euler characteristic. As a more recent application, we exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also compute various cohomology rings. (Joint work with S.Akbulut et al.)