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Unit sphere bundle equivalent to CP^{r-1}

Source: Miklós Schweitzer 2016, Problem 7

November 2, 2016

spheretopology

Problem Statement

Show that the unit sphere bundle of the

r

-fold direct sum of the tautological (universal) complex line bundle over the space

\mathbb{C}P^{\infty}

is homotopically equivalent to

\mathbb{C}P^{r-1}