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3 mistake in the formulae

It's a nice exercise in characteristic classes to show that S^4k for all k are NOT complex manifolds.

EDIT: I will answer Charlie's comment here and provide a sketch of the proof.

Let $\omega=TS^{4k}$ be the tangent space to the $4k$-sphere. If $S^{4k}$ was actually a complex manifold then $\omega$ would be a complex vector bundle. In this case the complexification of the underlying real vector bundle $\omega_{\mathbb{R}}$ would be canonically isomorphic to the Whitney sum $\omega\oplus \bar{\omega}$ (Milnor&Stasheff page 176). Now by corollary 15.5 in Milnor&Stasheff $$p_k(\omega_{\mathbb{R}})=c_k^2(\omega)-2c_{k-1}c_{k+1}(\omega)+\cdots\mp c_{2k}(\omega)$$2c_{2k}(\omega)$$This then shows that the top Pontrjagin number$$< p_k,[S^{4k}]>=<\mp c_{2k},[S^{4k}]>=\mp 2c_{2k},[S^{4k}]>=\mp 4$$but we also know that spheres are boundries of an oriented manifold and thus have higher Pontrjagin number 0. Contradiction. On another note, according to C.C. Hsiung's book Almost Complex and Complex Structures on page 233 he says "In fact, the absence of an almost complex structure on S^{4k} for k\geq 1 and S^{2n} for n\geq 4 was proved by Wu and jointly Borel and Serre respectively." 2 corrected spelling, capitalization It's a nice exercise in characteristic classes to show that S^4k for all k are NOT complex manifolds. EDIT: I will answer Charlie's comment here and provide a sketch of the proof. Let \omega=TS^{4k} be the tangent space to the 4k-sphere. If S^{4k} was actually a complex manifold then \omega would be a complex vector bundle. In this case the complexification of the underlying real vector bundle \omega_{\mathbb{R}} would be canonically isomorphic to the Whitney sum \omega\oplus \bar{\omega} (Milnor&Stasheff page 176). Now by corollary 15.5 in Milnor&Stasheff$$p_k(\omega_{\mathbb{R}})=c_k^2(\omega)-2c_{k-1}c_{k+1}(\omega)+\cdots\mp c_{2k}(\omega)$$This then shows that the top Pontrjagin number$$< p_k,[S^{4k}]>=<\mp c_{2k},[S^{4k}]>=\mp 4 but we also know that spheres are boundries of an oriented manifold and thus have higher Pontrjagin number 0. Contradiction.

On another note, according to C.C. Hsiung's book Almost Complex and Complex Structures on page 233 he says "In fact, the absence of an almost complex structure on $S^{4k}$ for $k\geq 1$ and $S^{2n}$ for $n\geq 4$ was proved by Wu and jointly Borel and Serre respectively."

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It's a nice exercise in characteristic classes to show that S^4k for all k are NOT complex manifolds.