We determine when there exists a ribbon rational homology cobordism between two connected sums of lens spaces, i.e. one without $3$-handles. In particular, we show that if space $L$ admits to different space, then must be homeomorphic $L(n,1)$, up orientation-reversal. As an application, classify $\chi$-concordances $2$-bridge links. Our work builds on Lisca's embeddings linear lattices.

Let X and Y be real normed spaces f : ? a surjective mapping. Then satisfies { ? ( x ) + y , ? } = ? if only is phase equivalent to linear isometry, that is, ? U where isometry 1 . This Wigner's type result for spaces.

‎in this paper first we define the morphism between geometric spaces in two different types‎. ‎we construct two categories of $uu$ and $l$ from geometric spaces then investigate some properties of the two categories‎, ‎for instance $uu$ is topological‎. ‎the relation between hypergroups and geometric spaces is studied‎. ‎by constructing the category $qh$ of $h_{v}$-groups we answer the question...

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