A branch of topology that deals with the notion of a fiber bundle.

learn more… | top users | synonyms

1
vote
0answers
10 views

Rational homology of $\Omega^{n+1}\Sigma^{n+1}X$

I want to know how compute, by induction and using the Serre spectral sequence for homology, $H_*(\Omega^{n+1}\Sigma^{n+1}X, \mathbb{Q})$. I know that I have to use the path-loop fibration $$ ...
2
votes
1answer
61 views

Local coefficients involved in the obstruction class for a lift of a map

I'm interested in understanding the importance of the local coefficients in the definition of the obstruction cocycle for a lift of a map $f\colon X \to B$ along a fibration $p \colon E \to B$. I'm ...
0
votes
0answers
19 views

Construction of Moore-Postnikov Tower

Given a map $f:X\to Y$, a Moore-Postnokiv tower is a sequence of spaces $...\to Z_3 \to Z_2 \to Z_1$ together with maps $X \to Z_n$ inducing isomorphism on $\pi_i$ for $i<n$ and surjection for ...
0
votes
1answer
33 views

Fiber bundle and fibration of classifying space

Let $BG$ is classifying space of $G$ topological group. If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the inclusion map $i:H\rightarrow G$ induces \begin{equation*} ...
5
votes
1answer
41 views

Fibration over contractible space is homotopic to a fiber

Let $\pi: E \to B$ be a fibration of $E$ over $B$, let $F = \pi^{-1}(b)$ for some $b \in B$ be a representative fiber, and suppose that $B$ is contractible. Is it always the case (or are there some ...
2
votes
1answer
40 views

Isotrivial family: different definitions

Let $f:X\to B$ a flat morphism of varieties over an algebraically closed field $k$. If $f$ is flat and with connected fibres we say that $f:X\to B$ is a family over $B$. In literature you can find ...
0
votes
0answers
18 views

Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
0
votes
1answer
24 views

Is any section an open map?

This "proof" of mine sounds deceivingly simple, and I think the result is too general to hold: also, I can't find any confirmation online. I would be grateful is somebody could have a look, and maybe ...
2
votes
1answer
61 views

Universal covering space VS fibration from contractible total space

For a path-connected space $X$, a covering space is a fiber bundle with a discrete set. It is known that if $X$ in addition locally path-connected and semilocally simply-connected, then $X$ has a ...
1
vote
0answers
27 views

reversal of constant rank theorem

Assuming $M$ is a manifold of dimension $\dim M = n+k$ and $F \colon M \to \mathbb{R}^n$ is a smooth function, such that the levelsets $F^{-1}({c})$ are submanifolds of dimension $k$. Can I therefore ...
7
votes
1answer
80 views

Fibration: if map $i^*: H^*(X, G) \to H^*(F, G)$ is surjective, then action of $\pi_1(B)$ on $H^*(F, G)$ trivial?

For a fibration $F \overset{i}{\to} X \overset{p}{\to} B$ with $B$ path-connected, if the map $i^*: H^*(X, G) \to H^*(F, G)$ is surjective, then does it necessarily follow that the action of ...
2
votes
0answers
39 views

Fibration induced by an almost complex structure

Let $E \rightarrow M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $$End(E)\rightarrow M,$$ denoted by ...
2
votes
1answer
50 views

Given a column vector, can we add columns, continuously dependent on the given one, to get an invertible matrix?

Given a vector $x$ in the $n=6$-dimensional Euclidian space $\mathbb{R}^n$, do there exist $n-1$ continuous functions $f_1$ to $f_{n-1}$ such that the matrix $$(x,f_1(x), … ,f_{n-1}(x))$$ is ...
1
vote
0answers
34 views

Fibration and induced mapping

I need help in such a problem, suppose it's rather simple one, but I have no skill in a field.. If $p: E\rightarrow B$ - fibration with a fibre F, then for every locally compact space $X$ an induced ...
4
votes
0answers
41 views

Why generalized vectors can be written locally as sum of vectors and 1-forms?

I would like to understand better this point. In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle ...
2
votes
1answer
48 views

How to construct the horizontal bundle?

I am learning the concept of connection. I am confused by the construction of the horizontal bundle. My question is: For a fibre bundle $M\rightarrow B$, the vertical vector space $V$ can be easily ...
1
vote
0answers
65 views

Is there a generalization of the Quaternionic Hopf fibrations and its natural connection?

For the quaternionic Hopf fibrations, $S^3\rightarrow S^7\rightarrow S^4$, we have a natural BPST connection form. Do we have some generalization of it? For example, the 'natural' connection form on ...
3
votes
1answer
80 views

examples for fibration not fibre bundle

We can use path space to make a map into a fibration. Generally, is this construction of fibration a fiber bundle? Or can someone give me some examples of fibration not fiber bundle?
10
votes
1answer
162 views

Gap between “fibration” and “fiber bundle”.

There are fibrations $E \rightarrow B$ which are not fiber bundles. Example: $E = [0,1]^2 / \text{middle vertical line segment}$ and $B=[0,1]$. In this example, $E$ has the homotopy type of a total ...
3
votes
0answers
111 views

Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
2
votes
0answers
23 views

Polynomials as Locally Isotrivial Covers

Let $k=\mathbb{A}^1$ be algebraically closed of arbitrary characteristic. I am interested in understanding when a polynomial $f:\mathbb{A}^n\to\mathbb{A}^1$ defines a locally isotrivial family over ...
1
vote
0answers
41 views

Bundle map is isomorphism iff it covers a homeomorphism

Consider $P_0$ and $P_1$ principal G-bundles with projection maps $\pi_0, \pi_1$, respectively; $f:P_0 \rightarrow P_1$ a continuous G-equivariant map (i.e. a bundle map) and $g:X_0 \rightarrow X_1$ ...
1
vote
2answers
63 views

Pullbacks and homotopy equivalences

Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it ...
4
votes
1answer
73 views

Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
1
vote
0answers
33 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
1
vote
0answers
67 views

the geometric realization of a simplicial set is contractible

Let $M$ be a monoid up to homotopy. The simplicial set $WM$ is defined by setting $$ WM_n=M^{n+1}=\{(g_0, g_1,\cdots,g_n)\mid g_i\in M\} $$ with faces and degeneracies given by \begin{eqnarray*} ...
2
votes
0answers
35 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
2
votes
1answer
78 views

question about “Homology fibrations and the group completion theorem”

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18: we have a fibre bundle $M_\infty\to (M_\infty)_M\to BM$ with $(M_\infty)_M$ constractible. In ...
0
votes
0answers
36 views

Fibrations over topoi

Let $\mathcal{S}$ be an elementary topos. What is (exactly) the relation between $\mathcal{S}$-indexed categories and fibrations over $\mathcal{S}$? Where can I read about this? (Or even find the ...
5
votes
1answer
171 views

Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is ...
2
votes
1answer
106 views

Getting fiber bundles from short exact sequences

Are there conditions that guarantee that a split short exact sequence of groups $$ 1 \rightarrow K \rightarrow G \rightarrow Q \rightarrow 1 $$ gives rise to a fiber bundle $$ F \rightarrow E ...
1
vote
1answer
98 views

Pull-back of a fibration along a homotopy equivalence

Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ and $g:B\rightarrow B'$ be homotopy inverses. Denote by ...
4
votes
1answer
80 views

Fibered product of Hopf Fibrations

I am wondering: what is the vector bundle associated to the Hopf-fibration $S^{3}\to S^{2}$? What is the fibered product of two Hopf-fibrations? Thanks.
1
vote
1answer
84 views

Rational map on $\mathbb P^1$ and its fibers

Consider a non-singular complex projective surface $S$ and a rational map $\psi:S\longrightarrow \mathbb P^1$; moreover suppose that $\psi$ is not defined on $\Delta=\{x_1,\ldots,x_m\}\subset S$. Now ...
2
votes
0answers
25 views

Two different notions of covering homotopy?

In Steenrod's The Topology of Fibre Bundles, there are two different notions of covering homotopy. One is furnished in Theorem 11.3: Let $\mathcal B,\mathcal B'$ be two bundles having the same ...
0
votes
0answers
37 views

Serre fibration and CW-complexes

Suppose that $p:X\rightarrow E$ is a Serre fibration. I know the definition: then $p$ has the right lifting property with respect to all inclusions $I^n\rightarrow I^{n}\times I$. Now it seems that ...
0
votes
1answer
47 views

Points with smooth fibers form an open subset

Let $S$ be a complex non-singular projective surface and let $C$ be a complex non-singular projective curve. Moreover consider a morphism $\varphi:S\longrightarrow C$ which is flat, proper and with ...
0
votes
0answers
31 views

Fibration with CW-complex as basespace admits retraction

Suppose $f:E\rightarrow B$ has the right lifting property with respect to all CW-pairs $(X,A)$. Then $f$ is a Serre fibration and also a weak-homotopy equivalence. But want i want to prove is the ...
1
vote
1answer
53 views

Fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
2
votes
0answers
39 views

Sequence of cofibrations gives strict commutativity of given triangle

The problem is the following: Suppose $X_0\rightarrow X_1\rightarrow\cdots$ is a sequence of cofibrations. We will denote $f_n:X_n\rightarrow X_{n+1}$. Assume that we have also maps ...
1
vote
1answer
95 views

Fiber bundles and manifolds

Each vector bundle is an example of a fibre bundle with some extra structure. This extra structure provides the algebraic object consisting of all sections (continuos or smooth) of given bundle. When ...
2
votes
1answer
103 views

Relative $CW$-complexes and Serre fibrations

Suppose we have a (continuous) map $p:E\rightarrow X$. Assume that $p$ has the right lifting property with respect to any relative $CW$-complex $i:A\rightarrow B$. I want to show that $p$ is a Serre ...
2
votes
2answers
88 views

Fibration $p_1:P(Y,y_0)\to Y$ has section iff $Y$ is contractible.

Let $P(Y,y_0) = \{ \omega : \omega(0) = y_0 \}$ be path space let's consider a fibration $p_1:P(Y,y_0)\to Y$ such that $\omega \mapsto \omega(1)$. Show that there exists $s: Y \to P(Y,y_0)$ such ...
3
votes
1answer
94 views

fibration of real projective space over complex projective space

From the fibration $$U(1)=S^1\to S^{2n+1}\to \mathbb{C}P^n,$$ can we quotient the action of $S^0$ and obtain a well-defined fibration $$ S^1/S^0=\mathbb{R}P^1\cong S^1\to ...
2
votes
1answer
87 views

fibration of complex projective space over quaternion projective space

From the fibration $$Sp(1)=S^3\to S^{4n+3}\to \mathbb{H}P^n,$$ can we quotient the action of $U(1)=S^1$ and obtain a well-defined fibration $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
3
votes
1answer
83 views

Postnikov towers for non-CW spaces

In the literature, Postnikov systems seem to be defined always in the setting of CW complexes. Looking at the proofs, it is not clear to me, why this assumption should be necessary. Question: Does ...
3
votes
3answers
136 views

$p:E\to B$ is fibration then $p_*:map(X,E)\to map(X,B)$ is fibration as well.

$p:E\to B$ is fibration then for $X$ being compactly generated weakly Hausdorff space $p_*:map(X,E)\to map(X,B)$ is fibration as well. We'd like to show that for any $Y$ and continuous $f$ and ...
1
vote
0answers
56 views

When does a homogeneous space define a fibration?

Let $G$ be a locally compact and $\sigma$-compact group acting continuously and transitively on locally compact Hausdorff $X$. Then if $x_0 \in X$ and $H_{x_0}$ denotes the isotropy group at $x_0$ we ...
2
votes
1answer
178 views

Being contractible in homotopy theory vs. homotopy type theory

I'm trying to clarify the notion of being contractible in homotopy theory vs. homotopy type theory. Is the following right? "In homotopy theory the real interval $[0,1]$, considered as a subset ...
7
votes
1answer
211 views

Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?