Even and odd graph homology (the commutative case).
Item
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Title
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Even and odd graph homology (the commutative case).
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Identifier
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AAI3144126
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identifier
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3144126
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Creator
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Nouri, Fereydoun.
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Contributor
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Adviser: Dennis Sullivan
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Date
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2004
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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In this paper we prove that odd commutative graph complex is quasi isomorphic to its subcomplex spanned by loop-less graphs (which is much smaller in size), provided the differential contracts only non-loop edges. In the case that the differential is allowed to contract loops as well as non-loop edges, we show that odd commutative graph homology vanishes in all dimensions. We also define the notion of an apple tree complex, and by analyzing the spectral sequence or its geometric realization we show that every apple tree complex is acyclic. We also observe that this spectral sequence contains rooted Lie trees as either the bottom or the top row of its E 1-term. From this new approach we relate the geometric realization of the Lie operad to the geometric realization of certain poset which is known to have a homotopy type of a wedge of spheres of appropriate dimension. Out of this we deduce a new proof that the homology of Lie operad is concentrated in the top degree. From the acyclicity of apple tree complexes, as a corollary, we deduce that even and odd commutative graph complexes are quasi-isomorphic with their quotients modulo the sub-complex of exactly 1-connected graphs. Even and odd graph homologies are calculated in low dimensions; various notions of orientations and their relations are discussed.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
