• Honam Mathematical Journal
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• v.21 no.1
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• pp.171-177
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• 1999

We find the properties of the defects and textures in an ordered medium. Especially, the space X, e.g., the residually solvable space or space satisfying the conditions ($T^{**}$) with respect to the defect and texture.

• Honam Mathematical Journal
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• v.24 no.1
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• pp.93-101
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• 2002

The $S^1$-Eule characteristics of X is defined by $\bar{\chi}_{S^1}(X)\;{\in}\;HH_1(ZG)$, where G is the fundamental group of connected finite $S^1$-compact manifold or connected finite $S^1$-finite complex X and $HH_1$ is the first Hochsch ild homology group functor. The purpose of this paper is to find several cases which the $S^1$-Euler characteristic has a homotopic invariant.