Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category

2017/03/10 by Rudyak, Yuli
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1703.03788

Abstract

Dranishnikov~\citeD2 proved that \rm cat X≤ \rm cd(π₁(X))+\lceil\frac\rm hd (X)-12\rceil. where \rm cd(π) denotes the cohomological dimension of a group π and \rm hd(X) denotes the homotopy dimension of X. Furthermore, there is a well-known inequality of Grossman,~\citeG: \rm cat X≤ \lceil\frac\rm hd (X)k+1\rceil if π_i(X)=0 for i≤ k. We make a synthesis and generalization of both of these results, by demonstrating the main result: \rm cat≤ \rm cd(π₁(X))+\lceil\frac\rm hd (X)-1k+1\rceil \text if π_i(X)=0 for i=2, …, k. The proof of the main theorem uses the Oprea--Strom inequality \rm cat X≤ \rm hd (Bπ₁(X))+\rm cat¹X, \citeOS where \rm cat¹ is the Clapp-Puppe \rm cat A with A the class of 1-dimensional CW complexes. The inequality clarified the Dranishnikov inequality.

Related

• The Lusternik-Schniremann-category and the fundamental group
• On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group
• Relating categorical dimensions in topology and symplectic geometry
• On the Coarse Lusternik-Schnirelmann Category of Groups
• The conjecture H: A lower bound of cohomologic dimension for an elliptic space