3
votes
1answer
86 views

Fibrations induced by deformation retractions

Given a map $f: A \to B$ and a homotopy equivalence $g: C \to A$, I wish to show that $E_f \to B$ and $E_{fg} \to B$ are fiber homotopy equivalent, where, given a function $f: A \to B$, $E_f$ is the ...
0
votes
1answer
25 views

Regarding an arbitary fibration as an inclusion

I'm reading through Allen Hatcher's Algebraic Topology, and he mentions that, given a Postnikov tower, the fibration $X_n \rightarrow X_{n-1}$, where $X_n$ and $X_{n-1}$ are CW complexes, can be ...
1
vote
1answer
49 views

Inverse limit of sequence of fibrations

I'm reading through the proof of Proposition 4.67 in Hatcher's Algebraic Topology, and I've come to something that I'm having trouble understanding. For an arbitrary sequence of fibrations $... ...
0
votes
0answers
36 views

about principal fibration

Supposing $\Omega C\rightarrow E\rightarrow B$ is a principal fibration and has a section, is $E$ fiber homotopy equivalent to $\Omega C\times B$? This is Exercise 4.3.22 of Hatcher's book. Thanks.
3
votes
0answers
329 views

Fibre and homotopy fibre

$p:E \rightarrow B$ is a fibration and $F$ is its fibre and $F_p$ its homotopy fibre. If $i:F \rightarrow F_p$ is the inclusion, is there a homotopy inverse $r$ of $i$ such that $r \circ i = id$?
1
vote
2answers
140 views

Does $\Omega$ functor preserve fibration?

If $p:E \rightarrow B$ is a fibration, is it necessarily that $\Omega p:\Omega E \rightarrow \Omega B$ a fibration? (where $\Omega E$ is the loopspace of $E$ and so is $\Omega B$)
3
votes
2answers
206 views

Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
5
votes
3answers
599 views

Showing that the loopspace $\Omega S^{\infty}$ is homotopic to $S^{\infty}$.

Showing that the infinite dimensional sphere $S^{\infty}$ is contractible is rather easy by constructing an explicit contraction (Hatcher gives a nice one). I thought it might be a nice exercise to ...
1
vote
1answer
120 views

Composition of fibrations

Suppose that $p:E \to B$ and $q:B \to B'$ are fibrations. Is it true that $qp:E \to B'$ is a fibration? I thought this might just be 'abstract-nonsense'. There is a diagram (possibly ...