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Abstract

In the first part of the work, we study the topological aspects of compact toric varieties. Considering toric varieties as pseudomanifolds, we investigate their standard stratification. We prove the triviality of the link bundles of toric varieties. On the other hand, we endow toric varieties with CW structure. We compute the homology groups of real 6-dimensional toric varieties with rational coefficients by using CW structure. We also determine the homology groups of the links of 4-, 6-, and 8-co-dimensional strata. At this point, we compare the topological description of toric varieties with the one from algebraic geometry. We show that we have similar characterizations of singular points in both pictures. At the end of part I, we introduce a class of pseudomanifolds called pseudo-toric varieties. We briefly study their link bundles and compute their Betti numbers. Finally, we generalize the notation of pseudomanifolds and introduce the so-called Q-pseudomanifolds. Showing that a Q-manifold with boundary satisfies the Lefschetz duality rationally gives us the necessary tools to generalize the theory of intersection spaces for Q-pseudomanifolds with isolated singularities. In the second part of the work, we study the theory of intersection spaces introduced by Banagl. We construct the intersection spaces of real 4-dimensional toric and pseudo-toric varieties and compute the associated Betti numbers. In the next step, we generalize the theory of intersection spaces to Q-pseudomanifolds. An arbitrary real 6-dimensional toric variety has stratification depth 2. But we can also consider it as a Q-pseudomanifold with isolated singularities. Applying the generalized construction of intersection spaces, we compute the associated Betti numbers of intersection spaces of 6-dimensional toric varieties. Comparing the Betti numbers of intersection spaces with the associated Betti numbers of intersection homology, we get the following results. Betti numbers of the intersection space are not combinatorial invariant. On the other hand, the intersection homology does not even determine the combinatorial data of the fan. In the last part of the work, we study the theory of intersection space pairs introduced by M. AGUSTÍN and J. FERNÁNDEZ DE BOBADILLA. As the first example, we endow real 6-dimensional toric varieties with the standard stratification and construct the associated intersection space pairs. Comparing our results with the last part yields that the Betti numbers of the intersection space pairs determine only the combinatorial data of the fan. We then construct the intersection space pairs of a specific class of 6-dimensional pseudo-toric varieties, where the generalization of the theory of intersection spaces is not applicable. Finally, we construct the intersection space of the link of an isolated singularity in an arbitrary 8-dimensional toric variety. Using the duality of the Betti numbers implies that, similar to real 6-dimensional toric varieties, we only have one non-combinatorial invariant parameter in the Betti numbers of the link.