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Abstract:
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The analysis of Grassmann manifolds, which were first introduced in the 19th century,
is one of the classical problems in the algebraic topology. When analyzing topological spaces, it is
always useful to determine their cohomology algebra. The cohomology of Grassmann manifolds
is already well known, but their covering spaces, so called oriented Grassmann manifolds, are
far less examined.
The oriented Grassmann manifold ˜Gn,k is defined to be the space of oriented k-dimensional
subspaces of Rn. In this dissertation we analyze the cohomology algebra of oriented Grassmann
manifolds ˜Gn,k with integer and modulo 2 coe!cients, predominantly the case k = 3. The
dissertation comprises three chapters. The first chapter is an introduction where an overview
of known results and necessary tools is given.
In the second chapter we study the cohomology with the modulo 2 coe!cients. First of all,
the known results in the case k = 2 are presented. Next, we move onto the case k = 3 where
the partial description of the cohomology algebra is given. This section is based on papers
published in the last several years. We give an overview of these results in the thesis, and we
also present original results for n close to a power of two. In the last part of this chapter, we
investigate the cohomology algebra of the manifold ˜G2t,4, and that is as far as we have come
with the examination of modulo 2 cohomology.
The third chapter is dedicated to the integral cohomology. This chapter, like the previous
one, also splits in several sections, depending on the value of k. When k = 2, the integral
cohomology is completely determined, and we present the proof for n odd. When k = 3, only
the integral cohomology of ˜Gn,3, n → {6, 8, 10}, has been determined so far, while for k ↭ 4
only some partial results are known. In this segment we also analyze the connection between
the integer and the modulo 2 cohomology algebra of these Grassmannians by analyzing the
morphism between them induced by the modulo 2 reduction. |