| 词条 | Hopf construction |
| 释义 |
In algebraic topology, the Hopf construction constructs a map from the join X*Y of two spaces X and Y to the suspension SZ of a space Z out of a map from X×Y to Z. It was introduced by {{harvs|txt|last=Hopf|authorlink=Heinz Hopf|year=1935}} in the case when X and Y are spheres. {{harvtxt|Whitehead|1942}} used it to define the J-homomorphism. ConstructionThe Hopf construction can be obtained as the composition of a map X*Y → S(X×Y) and the suspension S(X×Y) → S(Z) of the map from X×Y to Z. The map from X*Y to S(X×Y) can be obtained by regarding both sides as a quotient of X×Y×I where I is the unit interval. For X*Y one identifies (x,y,0) with (z,y,0) and (x,y,1) with (x,z,1), while for S(X×Y) one contracts all points of the form (x,y,0) to a point and also contracts all points of the form (x,y,1) to a point. So the map from X×Y×I to S(X×Y) factors through X*Y. References
|journal=Fund. Math.|volume= 25 |year=1935|pages= 427–440|url=http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv25i1p35bwm}}
1 : Algebraic topology |
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