• Nandini Nilakantan

      Articles written in Proceedings – Mathematical Sciences

    • Two-dimensional weak pseudomanifolds on eight vertices

      Basudeb Datta Nandini Nilakantan

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      We explicitly determine all the two-dimensional weak pseudomanifolds on 8 vertices. We prove that there are (up to isomorphism) exactly 95 such weak pseudomanifolds, 44 of which are combinatorial 2-manifolds. These 95 weak pseudomanifolds triangulate 16 topological spaces. As a consequence, we prove that there are exactly three 8-vertex two-dimensional orientable pseudomanifolds which allow degree three maps to the 4-vertex 2-sphere.

    • Homotopy type of neighborhood complexes of Kneser graphs, $K G_{2,k}$


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      Schrijver (Nieuw Archief voor Wiskunde, 26(3) (1978) 454–461) identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$ . Björner and de Longueville (Combinatorica 23(1) (2003) 23–34) proved that the neighborhood complex of the stable Kneser graph $SG_{n,k}$ is homotopy equivalent to a$k$-sphere. In this article, we prove that the homotopy type of the neighborhood complex of the Kneser graph $K G_{2,k}$ is a wedge of $(k + 4)(k + 1) + 1$ spheres of dimension $k$. We construct a maximal subgraph $S_{2,k}$ of $K G_{2,k}$ , whose neighborhood complex is homotopy equivalent to the neighborhood complex of $SG_{2,k}$ . Further, we prove that the neighborhood complex of $S_{2,k}$ deformation retracts onto the neighborhood complex of $SG_{2,k}$ .

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