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Hopf bundle (Definition)

Consider $ S^3\subset \mathbb{R}^4=\mathbb{C}^2$. The structure of $ \mathbb{C}^2$ gives a map $ \mathbb{C}^2-\{0\}\to\mathbb{C}P^1$, the complex projective line by the natural projection. Since $ \mathbb{C}P^1$ is homeomorphic to $ S^2$, by restriction to $ S^3$, we get a map $ \pi:S^3\to S^2$. We call this the Hopf bundle.

This is a principal $ S^1$-bundle, and a generator of $ \pi_3(S^2)$. From the long exact sequence of the bundle:

$\displaystyle \cdots\pi_n(S^1)\to \pi_n(S^3)\to\pi_n(S^2)\to\cdots$
we get that $ \pi_n(S^3)\cong \pi_n(S^2)$ for all $ n\geq 3$. In particular, $ \pi_3(S^2)\cong\pi_3(S^3)\cong\mathbb{Z}$.



"Hopf bundle" is owned by bwebste.
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Other names:  Hopf fibration
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Cross-references: generator, restriction, homeomorphic, projection, complex projective line, map, structure
There is 1 reference to this entry.

This is version 2 of Hopf bundle, born on 2002-12-27, modified 2002-12-27.
Object id is 3848, canonical name is HopfBundle.
Accessed 3579 times total.

Classification:
AMS MSC: 55R25 (Algebraic topology :: Fiber spaces and bundles :: Sphere bundles and vector bundles)
Pending Errata and Addenda
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Hopf Bundle by kudos on 2003-06-20 05:10:08
 H.Hopf (1895-1971) constructed the mapping (S^3 to S^2). His mapping pi is the most significant and simplified one.
 We can define S^1-action to S^3. The most remarkable fact (for Geometry) is that we can give the coordinates for our calculation to this Bundle as the example of Principal Fibre Bundle. 
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