Useful fibrations

Solution 1:

There's the path fibration $\Omega B \to PB \to B$, where for basepoint $* \in B$,

$PB = \{\gamma:[0,1]\to B \ |\ \gamma(0) = *\}$ is called the path space of $B$, and

$\Omega B = \{\gamma:[0,1]\to B \ |\ \gamma(0) = \gamma(1) = *\}$ is the loop space of $B$.

The map $p:PB \to B$ is the endpoint map $\gamma \mapsto \gamma(1)$.

I'm not quite sure what you mean by ``memorizing'' a fibration, but this is a useful one to understand.

Solution 2:

I recommend Introduction to Homotopy Theory, Arkowitz, Springer Universitext 2011, $\S$3.4, p.93, and Lecture Notes in Algebraic Topology, Davis & Kirk, AMS GSM 35 2002, $\S$4.3, 6.14, 7.7.

Let $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$ and $d=1,2,4=\dim_\mathbb{R}\mathbb{F}$.

$\bullet$ Let $n\!\in\!\mathbb{N}$. Let $\mathbb{S}^n_\mathbb{F}=\{(x_0,\ldots,x_n)\!\in\!\mathbb{F}^{n+1};\, \sum_{i=0}^n\!|x_i|^2\!=\!1\}=\mathbb{S}^{d(n+1)-1}$ and $\mathbb{P}_\mathbb{F}^n=\frac{\mathbb{F}^{n+1}\setminus\{0\}}{x\sim \lambda x;\, \lambda\in\mathbb{F}\setminus\{0\}}$. Then $$\mathbb{S}^{0}_\mathbb{F} \longrightarrow \mathbb{S}^n_\mathbb{F} \overset{p}{\longrightarrow} \mathbb{P}^{n}_\mathbb{F}$$ is a fiber bundle, the Hopf fibration, where $p(x_0,\ldots,x_n)\!=\!(x_0\!:\ldots:\!x_n)$. There holds $\mathbb{P}^1_\mathbb{F}=\mathbb{S}^d$.

$\bullet$ Let $0\!\leq\!l\!\leq\!k\!\leq\!n$. Let $V_\mathbb{F}^{k,n}$ $=$ $\{A\!\in\!\mathbb{F}^{n\times k};\, A\overline{A}^t\!=\!I_n\}$ $=$ $\{(v_1,\ldots,v_k)\!\in\!\mathbb{F}^{n\times k};\, v_1,\ldots,v_k\text{ are pairwise orthonormal}\}$ $\subseteq\mathbb{F}^{nk}$ with the subspace topology, the Stiefel manifold of real dimension $dnk\!-\!k\!-\!d\binom{k}{2}$, since $\|v_i\|\!=\!1$ gives $k$ equations over $\mathbb{R}$, and $v_iv_j\!=\!0$ for $i\!<\!j$ gives $\binom{k}{2}$ equations over $\mathbb{F}$. Then $$V^{k-l,n-l}_\mathbb{F} \!\!\longrightarrow V^{k,n}_\mathbb{F} \overset{p}{\longrightarrow} V^{l,n}_\mathbb{F}$$ is a fiber bundle, where $p(v_1,\ldots,v_k)\!=\!(v_{k-l+1},\ldots,v_k)$. There holds $V_\mathbb{F}^{1,n}\!\!=\!\mathbb{S}_\mathbb{F}^{n-1}$ and $V_\mathbb{R}^{n-1,n}\!\!\approx\!\mathrm{SO}^n, V_\mathbb{C}^{n-1,n}\!\!\approx\!\mathrm{SU}^n$ and $V_\mathbb{R}^{n,n}\!\approx\!\mathrm{O}^n, V_\mathbb{H}^{n,n}\!\approx\!\mathrm{U}^n, V_\mathbb{H}^{n,n}\!\approx\!\mathrm{Sp}^n$. Thus as a particular case of $l\!=\!1,k\!=\!n$ and $l\!=\!1,k\!=\!n\!-\!1$, we obtain fiber bundles $$\begin{array}{ccc} \mathrm{O}^{n-1}\longrightarrow\mathrm{O}^n\longrightarrow\mathbb{S}^{n-1}, &\mathrm{U}^{n-1}\longrightarrow\mathrm{U}^n\longrightarrow\mathbb{S}^{2n-1}, &\mathrm{Sp}^{n-1}\longrightarrow\mathrm{Sp}^n\longrightarrow \mathbb{S}^{4n-1},\\ \mathrm{SO}^{n-1}\longrightarrow\mathrm{SO}^n\longrightarrow\mathbb{S}^{n-1}, &\mathrm{SU}^{n-1}\longrightarrow\mathrm{SU}^n\longrightarrow\mathbb{S}^{2n-1}. &\\ \end{array}$$ $$\mathrm{SO}^n\longrightarrow\mathrm{O}^n\overset{\det}{\longrightarrow}\mathbb{S}^0,\hspace{8mm}\mathrm{SU}^n\longrightarrow\mathrm{U}^n\overset{\det}{\longrightarrow}\mathbb{S}^1$$

$\bullet$ Let $0\!\leq\!l\!\leq\!k\!\leq\!n$. Let $G_\mathbb{F}^{k,n}= \{k\text{-dimensional vector subspaces of }\mathbb{F}^n\}= \{V\!\leq\!\mathbb{F}^n;\, \dim_\mathbb{F}\!V\!=\!k\}$ with quotient topology obtained from the map $p\!:V_\mathbb{F}^{k,n}\!\rightarrow\!G_\mathbb{F}^{k,n}$ that sends $p(v_1,\ldots,v_k)\!=\!\langle v_1,\ldots,v_k\rangle$, the Grassmann manifold of real dimension ?. Then $$V^{k,k}_\mathbb{F} \longrightarrow V^{k,n}_\mathbb{F} \overset{p}{\longrightarrow} G^{k,n}_\mathbb{F}$$ is a fiber bundle, with $p$ given above. There holds $G^{1,n}_\mathbb{F}\!\!=\!\mathbb{P}_\mathbb{F}^{n-1}$, so Hopf fibrations are a special case.

$\bullet$ Let $n\!\in\!\mathbb{N}$ and $A\!\overset{\iota}{\longrightarrow}\!B\!\overset{p}{\longrightarrow}\!C$ be morphisms of groups (which are abelian if $n\!\geq\!2$) and $K(-,n)$ be the $n$-th Eilenberg–MacLane space functor. Then $0\!\longrightarrow\!A\!\overset{\iota}{\longrightarrow}\!B\!\overset{p}{\longrightarrow}\!C\!\longrightarrow\!0$ is an exact sequence of groups iff $$K(A,n)\: \overset{\iota_\ast}{\longrightarrow} \:K(B,n)\: \overset{p_\ast}{\longrightarrow} \:K(C,n)\text{ is a fibration.}$$

Solution 3:

The fibrations $O(n-1)\to O(n) \to S^{n-1}$, $U(n-1)\to U(n) \to S^{2n-1}$, and $Sp(n-1)\to Sp(n) \to S^{4n-1}$ from Bott periodicity are fairly important. Also mapping tori are fiber bundles. For example, the complement of a fibered knot in $S^3$.

Solution 4:

You mentioned this in your post, but there is also the fibration associated to any map of based spaces $f\colon (X,x_0)\to (Y,y_0)$ using the homotopy fiber: $\operatorname{hofib}(f)\to P_f\to Y$. Here, $P_f=\{(x,\gamma)\in X\times Y^I | \gamma(0)=f(x)\}$, and $\operatorname{hofib}(f)$ is the (strict) pullback of the maps $P_f\to Y$ with $(x,\gamma)\mapsto \gamma(1)$ and $*\to Y$ is the inclusion of the base point. Alternatively, it's the homotopy pullback of $X\to Y\leftarrow*$. The strict pullback of this diagram is $f^{-1}(y_0)$, which is why we are justified in calling this the homotopy fiber. Explicitly, $$\operatorname{hofib}(f)=\{(x,\gamma)\in X\times Y^I | \gamma(0)=f(x),\gamma(1)=y_0\}$$ Note that because $P_f$ is homotopy equivalent to $X$ (shrink the paths $\gamma$ to constant paths), this fibration is generally written $\operatorname{hofib}(f)\to X\to Y$. The main reason we like this fibration is because it gives us a long exact sequence of homotopy groups for any map of spaces.

This generalizes in the theory of cubical diagrams. Namely, given a map of cubical diagrams $Z\colon X\to Y$, we get a fibration with total fibers: $\operatorname{tfib}(Z)\to\operatorname{tfib}(X)\to\operatorname{tfib}(Y)$. The case in the above paragraph is the case where $Z$ is a map of $0$-cubes. A $0$-cube is a space, and the total fiber of a map of spaces is the homotopy fiber.