Abstract
We consider a q-analogue of abstract simplicial complexes, called q-complexes, and discuss the notion of shellability for such complexes. It is shown that q-complexes formed by independent subspaces of a q-matroid are shellable. Further, we explicitly determine the homology of q-complexes corresponding to uniform q-matroids. We also outline some partial results concerning the determination of homology of arbitrary shellable q-complexes.
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Notes
Recall that a topological space X is \(T_0\) (resp: \(T_1\)) if given any two distinct points of X, at least one of them (resp: each of them) is contained in an open set that does not contain the other point.
Indeed, a p-simplex is the convex hull of \(p+1\) points. So if \(p=-1\), then this is the empty set, while the singular p-simplex in X consists precisely of the empty function, and the free abelian group \(C_p(X)\) generated by it is \({\mathbb {Z}}\). On the other hand, all other chain complexes are 0.
Some of the computations in this example are done using SageMath, and the code is available at: drive.google.com/file/d/16_aAa2Us4maRx3rXA2wuaJZBQULCRxCc/view?usp=sharing
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We are grateful to the anonymous reviewers for many useful comments on a preliminary version of this paper.
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During the course of this work, Sudhir Ghorpade was partially supported by DST-RCN Grant INT/NOR/RCN/ICT/P-03/2018 from the Dept. of Science and Technology, Govt. of India, MATRICS grant MTR/2018/000369 from the Science and Engineering Research Board, and IRCC award Grant 12IRAWD009 from IIT Bombay.
During the course of this work, Rakhi Pratihar was supported by a doctoral fellowship at IIT Bombay from the University Grant Commission, Govt. of India (Sr. No. 2061641156). Currently, she is supported by Grant 280731 from the Research Council of Norway.
During the course of this work Tovohery Randrianarisoa was supported by a postdoc fellowship at IIT Bombay from the Swiss National Science Foundation Grant No. 181446.
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Ghorpade, S.R., Pratihar, R. & Randrianarisoa, T.H. Shellability and homology of q-complexes and q-matroids. J Algebr Comb 56, 1135–1162 (2022). https://doi.org/10.1007/s10801-022-01150-1
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DOI: https://doi.org/10.1007/s10801-022-01150-1