2
votes
1answer
109 views

Equivalence of the total spaces of two Serre fibrations with equivalent fibers

Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such ...
5
votes
2answers
238 views

Why is the path fibration a strong Hurewicz fibration?

In May and Sigurdsson "Parametrized homotopy theory" there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more ...
19
votes
3answers
611 views

Is the counit of geometric realization a Serre fibration?

Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the ...
0
votes
0answers
143 views

Change the fiber of a fibration

Let $F \rightarrow E \rightarrow B$ be a (Serre) fibration of topological spaces. Given a map $F' \rightarrow F$, is there a criterion for the existence or even an explicit construction of a fibration ...
3
votes
1answer
313 views

Free Loops, Moore Paths and the Borel Construction

My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homotopy equivalent ...
2
votes
2answers
507 views

Is every long exact sequence of homotopy groups induced by a fibration?

Is every long exact sequence $$\cdots\to\pi_{d+1}(B)\to\pi_d(F)\to\pi_d(E)\to\pi_d(B)\to\pi_{d-1}(F)\to\cdots$$ with topological spaces $F,E$ and $B$, where $F$ is a subspace of $E$ with inclusion map ...
3
votes
1answer
255 views

Terminology for fiberwise maps

I would like to know the standard terminology for the following two notions. Notion 1: $E_1\to B$ and $E_2\to B$ are fibrations over the same base space, and $f\colon E_1\to E_2$ is a map making the ...
10
votes
2answers
697 views

is there any fibration $\mathbb{R}^n\to \mathbb{S}^n$?

It is probably a trivial question. But I don't see the answer. Is there any Hurewicz fibration $\mathbb{R}^n\to \mathbb{S}^n$ ? Is there any fibration $X\to \mathbb{S}^n$, when $X\subset ...
3
votes
3answers
1k views

The fiber of a Serre fibration

If $p:E\to B$ is a Serre fibration (assume it is surjective), then for each $b\in B$ we get a comparison map $p^{-1}(b) \to F_b$, where $F_b$ is the homotopy fiber of $p$ over $b$. It is easy to ...
10
votes
6answers
793 views

A conceptual proof that local fibrations over paracompact spaces are global fibrations?

I'm teaching some homotopy theory at the moment, and I'm discovering a number of things I've never before learned properly. (I guess everybody who's ever taught knows that feeling.) One of those ...
1
vote
1answer
194 views

Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces

Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber. Question: How do you prove that the following diagram of homotopy groups commutes?: $\pi_n(Y) \to ...
2
votes
0answers
417 views

How to prove that a map is a Serre fibration?

I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that ...