The alternating sum of Betti numbers, which represent the rank of this type of structure, gives the topological definition of the Euler characteristic.
The alternating sum of Betti numbers, which represent the rank of this type of structure, gives the topological definition of the Euler characteristic. Showing that a topological space is simply connected is the same as showing that one of these structures is trivial. That one of these structures is defined on the given space as the set of all the equivalent paths that start and end at the same point and can be used to differentiate between the 2-sphere and the torus. Felix Klein asserted that the definition of geometry was the study of invariants of the "transformation" type of these structures in his Erlangen program. Adding inverse elements to monoids generates these structures. For 10 points, name these structures consisting of a single associative binaryoperation and a set that contains an identity and inverse elements, the simplest example of which are the integers under addition.
The alternating sum of Betti numbers, which represent the rank of this type of structure, gives the topological definition of the Euler characteristic.
groups [accept homology groups or homotopy groups or fundamental  groups prompt on monoid until "inverse"]