Editing Homotopy group (section)

== Long exact sequence of a fibration ==
Let <math>p : E \to B</math> be a basepoint-preserving [[Serre fibration]] with fiber <math>F,</math> that is, a map possessing the [[homotopy lifting property]] with respect to [[CW complex]]es. Suppose that ''B'' is path-connected. Then there is a long [[exact sequence]] of homotopy groups
<math display="block">\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots \to \pi_0(E) \to 0.</math>

Here the maps involving <math>\pi_0</math> are not [[group homomorphism]]s because the <math>\pi_0</math> are not groups, but they are exact in the sense that the [[Image (mathematics)|image]] equals the [[Kernel (algebra)|kernel]].

Example: the [[Hopf fibration]]. Let ''B'' equal <math>S^2</math> and ''E'' equal <math>S^3.</math> Let ''p'' be the [[Hopf fibration]], which has fiber <math>S^1.</math> From the long exact sequence
<math display="block">\cdots \to \pi_n(S^1) \to \pi_n(S^3) \to \pi_n(S^2) \to \pi_{n-1} (S^1) \to \cdots</math>

and the fact that <math>\pi_n(S^1) = 0</math> for <math>n \geq 2,</math> we find that <math>\pi_n(S^3) = \pi_n(S^2)</math> for <math>n \geq 3.</math> In particular, <math>\pi_3(S^2) = \pi_3(S^3) = \Z.</math>

In the case of a cover space, when the fiber is discrete, we have that <math>\pi_n(E)</math> is isomorphic to <math>\pi_n(B)</math> for <math>n > 1,</math> that <math>\pi_n(E)</math> embeds [[injective]]ly into <math>\pi_n(B)</math> for all positive <math>n,</math> and that the [[subgroup]] of <math>\pi_1(B)</math> that corresponds to the embedding of <math>\pi_1(E)</math> has cosets in [[bijection]] with the elements of the fiber.

When the fibration is the [[homotopy fiber|mapping fibre]], or dually, the cofibration is the [[mapping cone (topology)|mapping cone]], then the resulting exact (or dually, coexact) sequence is given by the [[Puppe sequence]].

=== Homogeneous spaces and spheres ===
There are many realizations of spheres as [[homogeneous space]]s, which provide good tools for computing homotopy groups of Lie groups, and the classification of principal bundles on spaces made out of spheres.

==== Special orthogonal group ====
There is a fibration<ref>
{{cite book
 | last1=Husemoller | first1=Dale
 | title=Fiber Bundles
 | series=Graduate Texts in Mathematics
 | volume=20
 | date=1994
 | publisher=Springer
 | doi=10.1007/978-1-4757-2261-1 | doi-access=free
 | page=89
 | isbn=978-1-4757-2263-5
}}</ref>
<math display="block">\mathrm{SO}(n-1) \to \mathrm{SO}(n) \to \mathrm{SO}(n) / \mathrm{SO}(n-1) \cong S^{n-1}</math>
giving the long exact sequence
<math display="block">\cdots \to \pi_i(\mathrm{SO}(n-1)) \to \pi_i(\mathrm{SO}(n)) \to \pi_i\left(S^{n-1}\right) \to \pi_{i-1}(\mathrm{SO}(n-1)) \to \cdots</math>
which computes the low order homotopy groups of <math>\pi_i(\mathrm{SO}(n-1)) \cong \pi_i(\mathrm{SO}(n))</math> for <math>i < n-1,</math> since <math>S^{n-1}</math> is <math>(n-2)</math>-connected. In particular, there is a fibration

<math display="block">\mathrm{SO}(3) \to \mathrm{SO}(4) \to S^3</math>
whose lower homotopy groups can be computed explicitly. Since <math>\mathrm{SO}(3) \cong \mathbb{RP}^3,</math> and there is the fibration
<math display="block">\Z/2 \to S^n \to \mathbb{RP}^n</math>
we have <math>\pi_i(\mathrm{SO}(3)) \cong \pi_i(S^3)</math> for <math>i > 1.</math> Using this, and the fact that <math>\pi_4\left(S^3\right) = \Z/2,</math> which can be computed using the [[Postnikov system]], we have the long exact sequence
<math display="block">\begin{align}
  \cdots \to{} &\pi_4(\mathrm{SO}(3)) \to \pi_4(\mathrm{SO}(4)) \to \pi_4(S^3) \to \\
         \to{} &\pi_3(\mathrm{SO}(3)) \to \pi_3(\mathrm{SO}(4)) \to \pi_3(S^3) \to \\
         \to{} &\pi_2(\mathrm{SO}(3)) \to \pi_2(\mathrm{SO}(4)) \to \pi_2(S^3) \to \cdots \\
\end{align}</math>

Since <math>\pi_2\left(S^3\right) = 0</math> we have <math>\pi_2(\mathrm{SO}(4)) = 0.</math> Also, the middle row gives <math>\pi_3(\mathrm{SO}(4)) \cong \Z\oplus\Z</math> since the connecting map <math>\pi_4\left(S^3\right) = \Z/2 \to \Z = \pi_3\left(\mathbb{RP}^3\right)</math> is trivial. Also, we can know <math>\pi_4(\mathrm{SO}(4))</math> has two-torsion.

===== Application to sphere bundles =====
Milnor<ref>{{cite journal|last=Milnor|first=John|date=1956|title=On manifolds homeomorphic to the 7-sphere|journal=Annals of Mathematics|volume=64|issue=2 |pages=399–405|doi=10.2307/1969983 |jstor=1969983 }}</ref> used the fact <math>\pi_3(\mathrm{SO}(4)) = \Z\oplus\Z</math> to classify 3-sphere bundles over <math>S^4,</math> in particular, he was able to find [[exotic sphere]]s which are [[smooth manifold]]s called [[Milnor's sphere|Milnor's spheres]] only homeomorphic to <math>S^7,</math> not [[diffeomorphic]]. Note that any sphere bundle can be constructed from a <math>4</math>-[[vector bundle]], which have structure group <math>\mathrm{SO}(4)</math> since <math>S^3</math> can have the structure of an [[Oriented manifold|oriented]] [[Riemannian manifold]].

=== Complex projective space ===
There is a fibration
<math display="block">S^1 \to S^{2n+1} \to \mathbb{CP}^n</math>
where <math>S^{2n+1}</math> is the unit sphere in <math>\Complex^{n+1}.</math> This sequence can be used to show the simple-connectedness of <math>\mathbb{CP}^n</math> for all <math>n.</math>