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A natural family of the Fredholm representations was constructed that lead to a symmetric vector bundle on completion of the fundamental group with a modification of the Higson-Roe corona. At the present talk we will discuss a homology version of symmetry in the case when completion with a modification of the Higson-Roe corona is a manifold with boundary. The results were developed by the discussion with N.Teleman during my visit in Ancona on March, 2007.