On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

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The following inequality X≤ Y+(X)-r/r+1 holds for every locally trivial fibration between ANE spaces f:X→ Y which admits a section and has the r-connected fiber where hd(X) is the homotopical dimension of X. We apply this inequality to prove that X≤ X-1/2+cd(π₁(X)) for every complex X with cd(π₁(X))≤ 2.