# A new lower bound for LS-category

Arabian Journal of Mathematics, Sep 2017

Let X be a simply connected CW-complex of finite type and $${\mathbb {K}}$$ an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of $$C_*(\Omega (X), \mathbb K)$$ to introduce a new homotopical invariant $$\textsc {r}(X, {\mathbb {K}})$$. If X is a Gorenstein space with nonzero evaluation map, then $$\textsc {r}(X, {\mathbb {K}})$$ turns out to interpolate $$\mathrm {depth}(H_*(\Omega (X), {\mathbb {K}}))$$ and $$\mathrm {e}_{{\mathbb {K}}}(X)$$. We also define for any minimal Sullivan algebra $$(\Lambda V,d)$$ a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant $$\textsc {r}(A,d)$$. When $$(\Lambda V,d)$$ is a minimal Sullivan model of X, this invariant fulfills the relation $$\textsc {r}(X, {\mathbb {K}}) = \textsc {r}(\Lambda V,d)$$.

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Youssef Rami. A new lower bound for LS-category, Arabian Journal of Mathematics, 2017, 1-9, DOI: 10.1007/s40065-017-0181-5