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Homotopy group

In mathematics, the first homotopy group of a pathwise-connected topological space \cal{M}, denoted by \pi_1(\cal{M}), is about the classification of all the loops in \cal{M}: The relations between loops that can be continuously deformed into one another is an equivalence relation. Then all loops in \cal{M} can be partitioned into disjoint equivalent classes. Finally, all equivalent classes of loops form a group.  Note: Let \alpha and \beta be two loops in \cal{M}, the group multiplication between the equivalent classes [\alpha] and [\beta] is defined as [\alpha]\cdot[\beta]=[\alpha * \beta], where \alpha * \beta is a concatenated loop that we first traverse through the loop \alpha and then through the loop \beta. In general, the n-th homotopy group, \pi_n(\cal{M}) is about the classification of the n-dimensional sphere S^n in \cal{M}. Examples: \pi_1\left(S^1\right)=\mathbb{Z} suggests that we can label each equivalent class of loops in $S^1...