Cartoon Based on the Following Papers

The fundamental group of a group system is  
	equal to the pushout of the group system
The fundamental group of a group system is equal to the pushout of the group system.

 

The second homotopy group of a group system is
	equal to the second homology group of the group system with
	twisted coefficients in the group ring of the fundamental group
	of the group system
The second homotopy group of a group system is equal to the second homology group of the group system with twisted coefficients in the group ring of the fundamental group of the group system.

 


The third homotopy group of a group system is equal to the 
third homology group of the tensor product over G of the 
group ring system with the second homotopy group of the group 
system with twisted coefficients.
The third homotopy group of a group system is equal to the third homology group of the tensor product over G of the group ring system with the second homotopy group of the group system with twisted coefficients.

 


The second homology group of the group system with twisted 
coefficients is isomorphic to the third homology group of the 
pair (Pushout of the group system, group system) with twisted 
coefficients.
The second homology group of the group system with twisted coefficients is isomorphic to the third homology group of the pair (Pushout of the group system, group system) with twisted coefficients.

 

The third homology group of the universal cover of a GEM complex
	is isomorphic to the third homotopy group of the group system of the GEM complex
	with twisted coefficients
The third homology group of the universal cover of a Generalized Eilenberg MacLane (GEM) complex is isomorphic to the third homotopy group of the group system of the GEM complex with twisted coefficients.

 

Oh, well ...
Oh, well ... !??