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18 votes
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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 ...
Bertram Arnold's user avatar
17 votes
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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 ...
Andy Putman's user avatar
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17 votes
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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 (...
Tyrone's user avatar
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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 ...
Mike Shulman's user avatar
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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 $\...
Mark Grant's user avatar
  • 35.4k
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 $Y$, ...
Mike Shulman's user avatar
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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 ...
Gael Meigniez's user avatar
12 votes
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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 ...
Alexander Betts's user avatar
11 votes
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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^...
Aleksandar Milivojević's user avatar
11 votes
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Shafarevich conjecture for abelian varieties

Let $B$ be a smooth projective curve over an algebraically closed field of characteristic zero. Let $K$ be the function field of $B$. Let $S$ be a finite set of closed points of $B$. You might find ...
Ariyan Javanpeykar's user avatar
10 votes

Where does the primary obstruction of a fibration show up in its spectral sequence?

In the general case of integral coefficients and possibly non-trivial local coefficient system, let $\pi=\pi_1(B)$. A cocycle for the obstruction class is an element $o\in Hom_{\mathbb Z\pi}(C_{k+1}\...
user95545's user avatar
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10 votes
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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.
Alexander Campbell's user avatar
9 votes
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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 ...
Dmitri Panov's user avatar
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9 votes
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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 ...
David Roberts's user avatar
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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 ...
Mike Shulman's user avatar
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8 votes
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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. ...
Nicholas Kuhn's user avatar
8 votes
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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 $C$ ...
Tom Goodwillie's user avatar
7 votes
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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 ...
Chris Schommer-Pries's user avatar
7 votes
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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 $(x,0)$...
Yonatan Harpaz's user avatar
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 ...
Mike Shulman's user avatar
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7 votes
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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-...
Tim Campion's user avatar
  • 62.3k
7 votes
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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/...
Mark Grant's user avatar
  • 35.4k
6 votes

Constructively, are all fibrations cloven?

If the axiom of choice holds then every fibration is cloven, and I would not be surprised if the converse holds. So this talk about not using the axiom of choice is best understood as closet-...
Andrej Bauer's user avatar
  • 47.8k
6 votes
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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 ...
Anton Fetisov's user avatar
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 ...
Robert Bruner's user avatar
6 votes
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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 ...
Will Sawin's user avatar
  • 139k
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 ...
Nicholas Kuhn's user avatar
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 $\Omega T\to F\to E$ ...
Mark Grant's user avatar
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6 votes
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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 $M$...
Thomas Rot's user avatar
  • 7,423
6 votes
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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 ...
Jeremy Brazas's user avatar

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