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Ethan Akin's ["proof"][2] that all vector bundles are stably trivial, and hence the $K$-theory of any space must vanish:

Let $V$ be a vector bundle over the base space $B$. Let $T$ be a trivial bundle of the same rank as $V$. To show that $V$ is stably trivial, it suffices to prove that $$V\oplus V=V\oplus T$$.

Let $P$ be the principal bundle associated with $V$. Pull $P$ back over itself to get a bundle $Q$:

[![Q defined as pullback of P against itself][1]][1]

Then $Q$ (together with the map to $B$) is the principal bundle associated to $V\oplus V$. But the bundle $Q\rightarrow P$ clearly has a section, namely the diagonal map (viewing $Q$ as a subspace of $P\times P$). Thus $Q=P\times GL_n$, which (together with the same map to $B$) is the principal bundle associated to $V\oplus T$.

(Reference: Ethan Akin, [K-theory doesn’t exist][2], JPAA 12 (1978) pp.177–179.) [1]: https://i.sstatic.net/BVYRc.gif [2]: http://www.sciencedirect.com/science/article/pii/0022404978900324

Ethan Akin's "proof" that all vector bundles are stably trivial, and hence the $K$-theory of any space must vanish:

Let $V$ be a vector bundle over the base space $B$. Let $T$ be a trivial bundle of the same rank as $V$. To show that $V$ is stably trivial, it suffices to prove that $$V\oplus V=V\oplus T$$.

Let $P$ be the principal bundle associated with $V$. Pull $P$ back over itself to get a bundle $Q$:

Then $Q$ (together with the map to $B$) is the principal bundle associated to $V\oplus V$. But the bundle $Q\rightarrow P$ clearly has a section, namely the diagonal map (viewing $Q$ as a subspace of $P\times P$). Thus $Q=P\times GL_n$, which (together with the same map to $B$) is the principal bundle associated to $V\oplus T$.

(Reference: Ethan Akin, K-theory doesn’t exist, JPAA 12 (1978) pp.177–179.)

Ethan Akin's "proof" that all vector bundles are stably trivial, and hence the $K$-theory of any space must vanish:

Let $V$ be a vector bundle over the base space $B$. Let $T$ be a trivial bundle of the same rank as $V$. To show that $V$ is stably trivial, it suffices to prove that $$V\oplus V=V\oplus T$$.

Let $P$ be the principal bundle associated with $V$. Pull $P$ back over itself to get a bundle $Q$:

Then $Q$ (together with the map to $B$) is the principal bundle associated to $V\oplus V$. But the bundle $Q\rightarrow P$ clearly has a section, namely the diagonal map (viewing $Q$ as a subspace of $P\times P$). Thus $Q=P\times GL_n$, which (together with the same map to $B$) is the principal bundle associated to $V\oplus T$.

Ethan Akin's ["proof"][2] that all vector bundles are stably trivial, and hence the $K$-theory of any space must vanish:

Let $V$ be a vector bundle over the base space $B$. Let $T$ be a trivial bundle of the same rank as $V$. To show that $V$ is stably trivial, it suffices to prove that $$V\oplus V=V\oplus T$$.

Let $P$ be the principal bundle associated with $V$. Pull $P$ back over itself to get a bundle $Q$:

[![Q defined as pullback of P against itself][1]][1]

Then $Q$ (together with the map to $B$) is the principal bundle associated to $V\oplus V$. But the bundle $Q\rightarrow P$ clearly has a section, namely the diagonal map (viewing $Q$ as a subspace of $P\times P$). Thus $Q=P\times GL_n$, which (together with the same map to $B$) is the principal bundle associated to $V\oplus T$.

(Reference: Ethan Akin, [K-theory doesn’t exist][2], JPAA 12 (1978) pp.177–179.) [1]: https://i.sstatic.net/BVYRc.gif [2]: http://www.sciencedirect.com/science/article/pii/0022404978900324

Ethan Akin's "proof" that all vector bundles are stably trivial, and hence the $K$-theory of any space must vanish:

Let $V$ be a vector bundle over the base space $B$. Let $T$ be a trivial bundle of the same rank as $V$. To show that $V$ is stably trivial, it suffices to prove that $$V\oplus V=V\oplus T$$.

Let $P$ be the principal bundle associated with $V$. Pull $P$ back over itself to get a bundle $Q$:

Then $Q$ (together with the map to $B$) is the principal bundle associated to $V\oplus V$. But the bundle $Q\rightarrow P$ clearly has a section, namely the diagonal map (viewing $Q$ as a subspace of $P\times P$). Thus $Q=P\times GL_n$, which (together with the same map to $B$) is the principal bundle associated to $V\oplus T$.

Ethan Akin's "proof" that all vector bundles are stably trivial, and hence the $K$-theory of any space must vanish:

Let $V$ be a vector bundle over the base space $B$. Let $T$ be a trivial bundle of the same rank as $V$. To show that $V$ is stably trivial, it suffices to prove that $$V\oplus V=V\oplus T$$.

Let $P$ be the principal bundle associated with $V$. Pull $P$ back over itself to get a bundle $Q$:

Then $Q$ (together with the map to $B$) is the principal bundle associated to $V\oplus V$. But the bundle $Q\rightarrow P$ clearly has a section, namely the diagonal map (viewing $Q$ as a subspace of $P\times P$). Thus $Q=P\times GL_n$, which (together with the same map to $B$) is the principal bundle associated to $V\oplus T$.