Whitehead Product
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High Quality Content by WIKIPEDIA articles! The Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in (Whitehead 1941). Given elements f in pi_k(X), g in pi_l(X), the Whitehead bracket [f,g] in pi_{k+l-1}(X) is defined as follows: The product S^k times S^l can be obtained by attaching a (k + l)-cell to the wedge sum S^k vee S^l; the attaching map is a map S^{k+l-1} to S^k vee S^l. Represent f and g by maps fcolon S^k to X and gcolon S^l to X, then compose their wedge with the attaching map, as S^{k+l-1} to S^k vee S...
High Quality Content by WIKIPEDIA articles! The Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in (Whitehead 1941). Given elements f in pi_k(X), g in pi_l(X), the Whitehead bracket [f,g] in pi_{k+l-1}(X) is defined as follows: The product S^k times S^l can be obtained by attaching a (k + l)-cell to the wedge sum S^k vee S^l; the attaching map is a map S^{k+l-1} to S^k vee S^l. Represent f and g by maps fcolon S^k to X and gcolon S^l to X, then compose their wedge with the attaching map, as S^{k+l-1} to S^k vee S^l to X The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of k + l 1(X).
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