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.)

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