19
votes
Accepted
Classifying space for fibrations with Eilenberg-MacLane space fibers and nontrivial fundamental group actions
Mark Grant's excellent answer already resolves the question. However, let me sketch how this arises as a special case of the more general problem of classifying fibrations with a given fiber.
For any ...
- 4,709
17
votes
Accepted
Foliation of $\mathbb R^n$ by connected compact manifolds
There does not, even if you don’t require the fiber and base to be manifolds (or even connected, just that $F$ is not a single point). See
Borel, Armand; Serre, Jean-Pierre,
Impossibilité de fibrer ...
- 44.8k
17
votes
Accepted
Are all homotopy equivalences realized by fibrations over [0,1]?
There is an old result due to Patricia Tulley which claims that this is possible.
P. Tulley, A strong homotopy equivalence and extensions for Hurewicz fibrations, Duke Math. J. 36(3): 609-619 (...
- 5,596
16
votes
Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?
I am answering your "later addon" only, although it seems actually to be a very different question than your original one.
This is perhaps one of the most misunderstood aspects of HoTT and ...
- 66.8k
15
votes
Classifying space for fibrations with Eilenberg-MacLane space fibers and nontrivial fundamental group actions
Denote $\pi=\pi_1(X)$ and fix the monodromy action $\rho:\pi\to \operatorname{Aut}(A)$. There is a generalized Eilenberg-Mac Lane space $L_\rho(A,n+1)$, whose only non-trivial homotopy groups are $\...
- 35.9k
14
votes
Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?
Here is an answer to your original question in the context of HoTT. An arbitrary map $f:X\to Y$ that isn't a fibration can't be viewed literally as a family of spaces varying continuously over $Y$, ...
- 66.8k
14
votes
Foliation of $\mathbb R^n$ by connected compact manifolds
On the other hand, if you only mean "foliation" as in your title, and not "fibration", then there is Vogt's foliation of R^3 by circles! (But it is not C^1, only differentiable).
Vogt, Elmar, "A ...
- 2,479
12
votes
Accepted
Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma
One of the many equivalent definitions of smoothness of a morphism $f\colon X\to Y$ of varieties over a field $k$ is that $f$ is smooth if and only if it is formally smooth. The latter means the ...
- 1,790
11
votes
Accepted
Obstructions for the lifting problem after a pull-back
First, note that since you are assuming $F$ is $d-1$-connected, the primary obstruction lies in $H^{d+1}$, not $H^d$.
Now, consider the diagram $\require{AMScd}$
\begin{CD}
& & & & S^...
- 1,471
10
votes
Accepted
A fibration equivalent to having a terminal object
A category $C$ has a terminal object if and only if the canonical functor $C \star \mathbf{1} \to \mathbf{1} \star \mathbf{1} = \mathbf{2}$ is a fibration.
- 5,539
10
votes
Connectivity of fibers under fibration replacement
No, this isn't true.
Fix $k \geq 1$ and let $Y$ be the join $S^k \ast [0,1]$, which is homeomorphic to $D^{k+2}$. It is a simply-connected CW-complex and has trivial homotopy groups.
Let $X$ be the ...
- 52.6k
9
votes
Accepted
Half-dimensional torus fibration vs Lagrangian torus fibration
This doesn't need to hold. For example, if one takes a $(T^4,\omega)$ with a constant symplectic structure $\omega$, in order for it to have a fibration by Lagrangian tori one should be able to find a ...
- 28.9k
9
votes
Accepted
Can we show that a functor is a fibration without choosing a cleavage?
Just as an example, given a category $\mathcal{C}$ with finite limits, showing $\mathrm{cod}\colon \mathcal{C}^\mathbf{2}\to \mathcal{C}$ is a fibration does not involve choosing a cleaving. All that ...
- 35.5k
9
votes
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Universal property of the codomain fibration
First a note about terminology: older literature sometimes uses the term "cofibration" for a functor $p:E\to B$ such that $p^{\rm op} : E^{\rm op} \to B^{\rm op}$ is a fibration, but it's ...
- 66.8k
8
votes
Accepted
Serre and Hurewicz fibrations definition for pointed spaces?
This all works as long as the basepoint $x_0$ of $X$ is nondegenerate. The general context for this is due to Arne Strom who showed that the category of topological spaces, with the classic (i.e. ...
- 11.1k
8
votes
Accepted
Associativity of consecutive fibrations
You are right: they are not equivalent. For an example, choose a group $G$ that has a normal subgroup $H$ in which there is a subgroup $K$ that is normal in $H$ but not in $G$. Take $A$, $B$, and $C$ ...
- 55.9k
7
votes
Accepted
Does the Eilenberg Moore Construction Preserve fibrations?
Let $C$ be a 2-category and $Mnd(C)$ the 2-category of monads in $C$. As explained by Street in the formal theory of monads, the Eilenberg-Moore construction is right 2-adjoint to the inclusion 2-...
- 63.9k
7
votes
Multiplicativity of Euler characteristic for non-orientable fibrations
Since this question seems to be attracting some renewed interest, I may as well point out that a few years after I asked it, Kate Ponto and I proved a generalization of this formula by purely ...
- 66.8k
7
votes
Accepted
About fibrations with fibre Eilenberg-MacLane spaces
No. If this were the case then there would be a section $s: B \to E$ to $f$ induced by the $G$-equivariant map $\widetilde{s}:\widetilde{B} \to \widetilde{B} \times {\rm K}(M,n)$ sending $x$ to $(x,0)$...
- 9,481
7
votes
Accepted
Is the fiberwise suspension of a Serre fibration a Serre fibration?
Lemma 6 in Section 3 of
Vandembroucq, Lucile, Fibrewise suspension and Lusternik-Schnirelmann category, Topology 41, No. 6, 1239-1258 (2002). ZBL1009.55002.
(also available at https://core.ac.uk/...
- 35.9k
7
votes
Accepted
Fibrations of categories with terminal objects
If all of the fibres in a fibration have terminal objects then they are automatically preserved by re-indexing (also called cartesian, horizontal or prone liftings) --- you don't need to state this as ...
- 8,481
6
votes
Accepted
Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?
Yes, there is a certain sense in which your statements are true. As Mike Shulman and Qiaochu Yuan said, the strict fiber of a map cannot be defined in HoTT and doesn't make sense, but you can work ...
- 4,780
6
votes
Accepted
Space of sections of a fibration under weak homotopy equivalence
This is not true in general, unless you assume the base is sufficiently nice (eg a CW-complex). Here is a counter-example.
Let $B = \mathbb{Q}$, the rationals with its topology as a subspace of the ...
- 27.5k
6
votes
Accepted
elliptic fibration over $\mathbb{P}^1$ with exactly two fibres with monodromy of unipotency rank 1
If you look at the global monodromy action $\pi_1 ( \mathbb P^1 -\{0,1\}) \to SL_2(\mathbb Z)$, you see that $\pi_1 ( \mathbb P^1 -\{0,1\}) = \mathbb Z$ so the image is abelian and therefore has ...
- 148k
6
votes
Trivialization of Pontryagin square on oriented $4$-manifolds
The original question was whether $\mathcal{P}(x)$ is non-zero when $x$ is. This answers that question, and was aimed at generic $r$, $p$ and $K$. Special cases may have special answers.
By Brown ...
- 3,526
6
votes
When is the cohomology of a fiber bundle a tensor product?
Lets analyze this when $\pi_1$ is finite, and the bundle is associated to a principal $\pi_1$-bundle, and we are considering rational coefficients.
Very generally, if a finite group $G$ acts freely ...
- 11.1k
6
votes
Delooping a fibration sequence with loopspace fiber and finite CW complexes
This question is addressed in the paper
Ganea, T., Induced fibrations and cofibrations, Trans. Am. Math. Soc. 127, 442-459 (1967). ZBL0149.40901.
A first observation is that $\Omega T\to F\to E$ ...
- 35.9k
6
votes
Accepted
CW structure on infinite-dimensional manifolds
Let $M$ be a Hilbert manifold. Then there exists a Riemannian metric $g$ on $M$ such that the induced metric $d$ is complete (See https://arxiv.org/pdf/1610.01527.pdf by Biliotti and Mercuri). Thus $M$...
- 7,583
6
votes
Accepted
Homotopy equivalent fibers and Fibrations
The figure below gives a simple but extreme counterexample, which I think has all the lifting properties one might want except for actually being a true fibration. The map is the identity everywhere ...
- 7,193
6
votes
Accepted
Action of fundamental group on homotopy fiber
(This answer is written in a model-independent fashion -- translate to your favourite formalism).
For every path $\gamma:[0,1]\to B$ you get an isomorphism in the homotopy category $X_{\gamma0}\...
- 16.5k
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