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Let$X$be a smooth$n$-dimensional projective variety embedded in some projective space ℙ^{$N$}over the field ℂ of the complex numbers. Associated with the general projection of$X$to a space ℙ^{$N$-$m$}($N$-$m$>$n$+1) one defines an extended Gauss map $\overline{\gamma}\colon\overline{X}\rightarrow\text{Gr}(n;N-m)$ (in case$N$-$m$>2$n$-1 this is the Gauss map of the image of$X$under the projection). We prove that $\overline{X}$ is smooth. In case any two different points of$X$do have disjoint tangent spaces then we prove that $\overline{\gamma}$ is injective.

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We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra $K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})$ for the algebraic$K$-theory of topological$K$-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the$K$-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective $A_\infty$ ring spectrum$R$with the Quillen$K$-theory of the abelian category of finitely generated $\pi_{0}R$ -modules.

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