This series of lectures aims to discuss the obstructions to the existence of almost complex structures. More precisely, we will consider the following two facts:

• The only spheres admitting an almost complex structure are $\Bbb S^2$ and $\Bbb S^6$, utilizing the Bott periodicity theorem in complex topological $K$-theory.
• Wu's theorem: A closed, oriented smooth $4$-manifold $M$ has an almost complex structure if and only if there exists $c \in H^2(M;\Bbb Z)$ such that $w_2(M) \equiv c \bmod 2$ and $\int_M c^2 = 2\chi(M) + 3\sigma(M)$, where $w_2$, $\chi$, and $\sigma$ denote the second Stiefel-Whitney class, the Euler characteristic, and the signature, respectively.

In the initial lectures, we will set the stage and cover all necessary prerequisites.


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