The third partial cohomology group and existence of extensions of semilattices of groups by groups

We introduce the concept of a partial abstract kernel associated to a group G and a semilattice of groups A and relate the partial cohomology group {H^{3}(G,C(A))} with the obstructions to the existence of admissible extensions of A by G which realize the given abstract kernel. We also show that if such extensions exist, then they are classified by {H^{2}(G,C(A))}.

On the classification of Schreier extensions of monoids with non-abelian kernel

We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}}. If an abstract kernel factors through {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}, where {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group {U(Z(A))} of invertible elements of the center {Z(A)} of A, on which M acts via Φ. An abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}), when it is not empty, is in bijection with the second cohomology group of M with coefficients in {U(Z(A))}.

COMPACT LIE GROUP ACTIONS ON CLOSED MANIFOLDS OF NON-POSITIVE CURVATURE

Borel proved that, if a finite group F acts effectively and continuously on a closed aspherical manifold M with centerless fundamental group π1(M), then a natural homomorphism ψ from F to the outer automorphism group Out π1(M) of π1(M), called the associated abstract kernel, is a monomorphism. In this paper, we investigate to what extent Borel's theorem holds for a compact Lie group G acting effectively and smoothly on a particular orientable aspherical manifold N admitting a Riemannian metric g0 of non-positive curvature in case that π1(N) has a non-trivial center. It turns out that if G attains the maximal dimension equal to the rank of Center π1(N) and the metric g0 is real analytic, then any element of G defining a diffemorphism homotopic to the identity of N must be contained in the identity component G0 of G. Moreover, if the inner automorphism group of π1(N) is torsion free, then the associated abstract kernel ψ : G/G0 → Out π1(N) is a monomorphism. The same result holds for the non-orientable N's under certain technical assumptions. Our result is an application of a theorem by Schoen–Yau [12] on harmonic mappings.