We will also show that although their fundamental groups are solvable, the infranilmanifolds we consider are in general not solvmanifolds, and hence they cannot be treated using the techniques developed for solvmanifolds.
Now you will want to get theorems of the following sorts: if X can be continuously deformed into another space Y, then their fundamental groups are isomorphic; the fundamental group of a product is the product of the fundamental groups; etc.
The algebraic fundamental group is best described as π 1 ( X, a ) = Z ^ ( 1 ) n ⋊ Gal ( k ).
end{aligned}Look now at the Albanese variety ({{mathrm{text {Alb}}}}(W)) of the Kähler manifold W, whose fundamental group is the quotient of the Abelianization of (Gamma = pi _1 (Z)) by its torsion subgroup.
This means that, if the genus of (X = C') is equal to (g'), then the orbifold fundamental group is isomorphic to the abstract group begin{aligned}&pi _{g',m_1, ldots,m_dd} :=langle alpha _1, beta _1, ldots, alpha _{g'},beta _{g'}, gamma _1, ldots, gamma _d | Pi _1^d gamma _j Pi _1^{g'} [ alpha _i, beta _i] = 1,&gamma _1^{m_1} = cdots = gamma _d^{m_d} =1 rangle.
However, any such dependence has to be continuous and the fundamental group is discrete, so in fact (gamma ) is independent of the (u_j) and, since (gamma ) is the identity when the (u_j) are in (T_+) (lifted to U), the associativity follows.
In fact there is a bijection between homotopy classes of self maps of Z and homomorphisms of (pi ), taken of course up to inner conjugation (inner conjugation is the effect of changing the base point, and if we do not insist on taking pointed spaces, i.e., pairs ((Z,z_0)) the action of a continuous map on the fundamental group is only determined up to inner conjugation).
The notion of digital fundamental group was originated by Khalimsky [E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proc. IEEE Int. Conf. Syst. Man Cybernet. (1987) 227 234].
For example, elements of the fundamental group are represented by loops.
A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers Analysis situs where the related concepts of homology and the fundamental group were also introduced.
