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Loop space homology of spaces of small category


Authors: Yves Félix and Jean-Claude Thomas
Journal: Trans. Amer. Math. Soc. 338 (1993), 711-721
MSC: Primary 55P35; Secondary 16E10, 55P62, 57T25
DOI: https://doi.org/10.1090/S0002-9947-1993-1134757-2
MathSciNet review: 1134757
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Abstract: Only little is known concerning $ {H_\ast}(\Omega X;{\mathbf{k}})$, the loop space homology of a finite $ {\text{CW}}$ complex $ X$ with coefficients in a field $ {\mathbf{k}}$. A space $ X$ is called an $ r$-cone if there exists a filtration $ \ast = {X_0} \subset {X_1} \subset \cdots \subset {X_r} = X$, such that $ {X_i}$ has the homotopy type of the cofibre of a map from a wedge of sphere into $ {X_{i - 1}}$. Denote by $ {A_X}$ the sub-Hopf algebra image of $ {H_\ast}(\Omega {X_1})$. We prove then that for a graded $ r$-cone, $ r \leq 3$, there exists an isomorphism $ {A_X} \otimes T(U)\xrightarrow{ \cong }{H_\ast}(\Omega X)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1134757-2
Keywords: Loop space, $ r$-cones, Lusternik-Schnirelmann category, Hopf algebra, global dimension, depth
Article copyright: © Copyright 1993 American Mathematical Society

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