The Hopf construction can be obtained as the composition of a map
and the suspension
of the map from to .
The map from to can be obtained by regarding both sides as a quotient of where is the unit interval. For one identifies with and with , while for one contracts all points of the form to a point and also contracts all points of the form to a point. So the map from to factors through .
Referencesedit
Hopf, H. (1935), "Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension", Fund. Math., 25: 427–440
Whitehead, George W. (1942), "On the homotopy groups of spheres and rotation groups", Annals of Mathematics, Second Series, 43 (4): 634–640, doi:10.2307/1968956, ISSN 0003-486X, JSTOR 1968956, MR 0007107
January 01, 1970
hopf, construction, algebraic, topology, constructs, from, join, displaystyle, spaces, displaystyle, displaystyle, suspension, displaystyle, space, displaystyle, from, displaystyle, times, displaystyle, introduced, hopf, 1935, case, when, displaystyle, display. In algebraic topology the Hopf construction constructs a map from the join X Y displaystyle X Y of two spaces X displaystyle X and Y displaystyle Y to the suspension S Z displaystyle SZ of a space Z displaystyle Z out of a map from X Y displaystyle X times Y to Z displaystyle Z It was introduced by Hopf 1935 in the case when X displaystyle X and Y displaystyle Y are spheres Whitehead 1942 used it to define the J homomorphism Construction editThe Hopf construction can be obtained as the composition of a map X Y S X Y displaystyle X Y rightarrow S X times Y nbsp and the suspension S X Y S Z displaystyle S X times Y rightarrow SZ nbsp of the map from X Y displaystyle X times Y nbsp to Z displaystyle Z nbsp The map from X Y displaystyle X Y nbsp to S X Y displaystyle S X times Y nbsp can be obtained by regarding both sides as a quotient of X Y I displaystyle X times Y times I nbsp where I displaystyle I nbsp is the unit interval For X Y displaystyle X Y nbsp one identifies x y 0 displaystyle x y 0 nbsp with z y 0 displaystyle z y 0 nbsp and x y 1 displaystyle x y 1 nbsp with x z 1 displaystyle x z 1 nbsp while for S X Y displaystyle S X times Y nbsp one contracts all points of the form x y 0 displaystyle x y 0 nbsp to a point and also contracts all points of the form x y 1 displaystyle x y 1 nbsp to a point So the map from X Y I displaystyle X times Y times I nbsp to S X Y displaystyle S X times Y nbsp factors through X Y displaystyle X Y nbsp References editHopf H 1935 Uber die Abbildungen von Spharen auf Sphare niedrigerer Dimension Fund Math 25 427 440 Whitehead George W 1942 On the homotopy groups of spheres and rotation groups Annals of Mathematics Second Series 43 4 634 640 doi 10 2307 1968956 ISSN 0003 486X JSTOR 1968956 MR 0007107 Retrieved from https en wikipedia org w index php title Hopf construction amp oldid 1195132669, wikipedia, wiki, book, books, library,