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Are causally isomorphic spacetimes Wick-related?

Take the time-orientable spacetimes $(M_1,g_1)$ and $(M_2,g_2)$. The Lorentzian metrics $g_1$ and $g_2$ induce binary relations $\ll_1$ and $\ll_2$ over their manifolds respectively which is defined ...
Bastam Tajik's user avatar
2 votes
0 answers
89 views

lifting a family of curves to a family of sections of a vector bundle?

This is a question in obstruction theory. It should be basic but I can't find a reference. Let's stick to the $C^\infty$ category, so all objects mentioned are smooth. Let $\pi: E \to M$ be a vector ...
skwok's user avatar
  • 21
4 votes
0 answers
168 views

Obstruction to finding a Whitney disk

Let $p,q\geq 3$ be integers. Let $M$ be a compact oriented smooth $(p+q)$-manifold and $P$ and $Q$ compact submanifolds of dimensions $p$ and $q$ intersecting transversely. Assume that $M,P$ and $Q$ ...
João Lobo Fernandes's user avatar
0 votes
1 answer
134 views

Vector bundles over a homotopy-equivalent fibration

I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here. Let $\pi:N\rightarrow M$ be a smooth ...
Bence Racskó's user avatar
14 votes
2 answers
1k views

Is the Gödel universe Wick rotatable?

Take Wick rotatability being as the way defined in the following article by Helleland and Hervik: Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
Bastam Tajik's user avatar
2 votes
0 answers
348 views

Is a Wick rotatable spacetime necessarily strongly causal?

There are a few viable ways to formulate Wick rotatability that preserve distinct features. One is mentioned in the post: Obtain Lorentzian manifolds from Riemannian ones by Wick rotation There's also ...
Bastam Tajik's user avatar
4 votes
1 answer
96 views

Some questions about the definition of Chern classes in Cheeger--Simons differential characters

In page 62 to 63 of the paper "Differential characters and geometric invariants" by Cheeger and Simons, they define, among other things, Chern classes taking values in differential ...
Ho Man-Ho's user avatar
  • 1,117
1 vote
0 answers
80 views

Gerstenhaber bracket for Hochschild cohomology with values in a module

I am currently trying to compute obstructions in a Hochschild cohomology $\mathrm{HH}^* (A,M)$ where $A$ is a $\Bbbk$-algebra and $M$ an $A$-bimodule. The obstruction I am looking at looks a lot like ...
Felix's user avatar
  • 203
2 votes
1 answer
141 views

Obstruction to a cohomology class on total space being a pullback of a class on the base space is the restriction to the fiber

Let $\pi \colon E \to X$ be a fiber bundle with fiber $F$ and suppose that $\tilde H^i(F) = 0$ for $0 \leq i \leq k-1$. Using the Leray-Serre spectral sequence, we get an exact sequence $$ 0 \to H^k(...
Motmot's user avatar
  • 293
9 votes
1 answer
296 views

Non-triviality of a Postnikov class in $H^3\left(B \operatorname{PSU}(N) ; \mathbb{Z}_q\right)$

Let $\alpha\in H^2(B\operatorname{PSU}(N) ; \mathbb{Z}_N)$ be the obstruction class for lifting a $\operatorname{PSU}(N)$-bundle to an $\mathrm{SU}(N)$-bundle. Note that $\operatorname{PSU}(N)\cong \...
Satoshi  Nawata's user avatar
0 votes
0 answers
61 views

Obstruction to finding a framing for quotient manifolds

The question is rather open-ended but I hope it is concrete enough. If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what ...
João Lobo Fernandes's user avatar
7 votes
2 answers
471 views

Injectivity of the cohomology map induced by some projection map

Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence $$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$ where $G_c$ is the normal subgroup which ...
Ye Weicheng's user avatar
15 votes
1 answer
1k views

Possible mistake in Cohen notes "Immersions of manifolds and homotopy theory" (version 27 March 2022)

In Theorem 2 of these notes, Ralph Cohen reformulates the main theorem of Hirsch-Smale theory merely in terms of normal bundles. In particular, he says that if $N, M$ are two manifolds, $\dim N< ...
Overflowian's user avatar
  • 2,523
8 votes
1 answer
412 views

Finite domination and compact ENRs

Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only ...
John Klein's user avatar
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10 votes
2 answers
388 views

Multiplicative structures on truncated Moore spectra

As discussed for example in this paper (and this MO thread), there are obstructions for the existence of $A_n$-structures on Moore spectra $M(p^r):=\mathbb{S}/p^r$, which in general don't vanish. In ...
KotelKanim's user avatar
  • 2,280
3 votes
0 answers
192 views

Virtual fundamental class of punctual Hilbert scheme of points

$\DeclareMathOperator\Hilb{Hilb}$It is well known that the Hilbert scheme $\Hilb^n(\mathbb C^3)$ has a (symmetric) perfect obstruction theory. Consider the punctual part at $0 \in \mathbb C^3$, which ...
user147163's user avatar
1 vote
0 answers
137 views

Obstruction to lifting homomorphism of groups

Is there a "cohomology" group that encodes obstructions to constructing a lift in a diagram of groups below? If $X\to Y$ is an extension and the bottom row is the identity map it's just $H^1(...
Grisha Taroyan's user avatar
9 votes
1 answer
314 views

Framed version of the "copants bordism"?

The "pants" bordism in dimension n is a bordism which goes from $S^n \sqcup S^n$ to $S^n$ witnessing the connected sum operation - equivalently by attaching 1-handle to the trivial bordism, ...
Chris Schommer-Pries's user avatar
1 vote
0 answers
71 views

Would the iterated finite abelian descent obstruction equality hold for curves?

Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, ...
oleout's user avatar
  • 875
1 vote
0 answers
83 views

Obstruction to deformation of composite morphism (Reference request + question)

Let $f_0:X_0\xrightarrow{g_0}Y_0\xrightarrow{h_0}Z_0/S_0$ be a morphism of smooth projective $S_0$-schemes such that $g_0,h_0$ are flat. Let $S_0\subset S$ be a first-order thickening, and let $X,Y,Z$ ...
pozio's user avatar
  • 599
3 votes
1 answer
129 views

Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...
iou's user avatar
  • 223
3 votes
1 answer
208 views

Definition of 1st degree obstruction class

Recently I go through obstruction class illustrated by Milnor. He defined $\mathfrak{o}_i$by an element in $H^i(M; \pi_{i-1}(V_{n-i+1}(F))$, which is cohomology with local coefficients. But the 0th ...
XT Chen's user avatar
  • 1,094
4 votes
0 answers
286 views

Where can I read about non-principal obstruction theory?

Most treatments of obstruction theory assume a principal Postnikov tower. I have the vague sense that if one uses cohomology with local coefficients, one does not need to make any assumptions on one's ...
Tim Campion's user avatar
  • 62.2k
2 votes
0 answers
169 views

obstruction cocycle for nonsimple spaces using local coefficients

This question is similar to here but I was hoping for a concrete theorem statement surrounding the obstruction cocycle for non-simple spaces. I'm hoping for a theorem like the following: Let $A \...
Andres Mejia's user avatar
3 votes
1 answer
349 views

Homotopy class of maps into Stiefel manifolds

Motivation Hopf theorem, asserts that $C^0$-maps $f:M^n\to \mathbb{S}^n$ from an orientable, closed n-manifold into an n-sphere are classified up to homotopy by their degree $deg(f)$. The theorem ...
Overflowian's user avatar
  • 2,523
2 votes
2 answers
211 views

Measuring failure of a setup to preserve some structure giving interesting notions

I am looking for some examples of failure of some structures giving interesting notions. For example, we have the following situation: Let $P(M,G)$ be a principal bundle. Let $\Gamma\subseteq TP$ be ...
7 votes
1 answer
670 views

Nowhere vanishing section implies reduction of structure group

Description I noticed a repeating theme in vector bundle theory, and wonder if there's a theorem that describes this kind of phenomenon. Given a vector bundle $E$ over a manifold $X$. If there is a ...
Student's user avatar
  • 5,038
8 votes
1 answer
566 views

Obstruction to homotopy, cohomology operations and Dold-Whitney theorem

I am reading the famous paper by Dold and Whitney "Classification of Oriented Sphere Bundles Over A 4-Complex". I'll state their theorem for the case of SO(3) bundles Classification Theorem:Let $B_1,...
Overflowian's user avatar
  • 2,523
11 votes
1 answer
810 views

What is the relationship between spectral sequences and obstruction theory?

Let $X,Y$ be objects of some category $\mathcal C$, and suppose I want to study homotopy classes of maps from $X$ to $Y$ (almost everything one does in algebraic topology can be viewed this way). It ...
Tim Campion's user avatar
  • 62.2k
11 votes
1 answer
1k views

Classification of bundles, Postnikov towers, obstruction theory, local coefficients

RECAP on classification of bundles We want to classify $G$-principal bundles over $X$ (smooth manifold, G compact Lie). These are in 1-1 correspondence with homotopy classes of maps $[X,BG]$ (where $...
Overflowian's user avatar
  • 2,523
5 votes
0 answers
684 views

Questions about obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
Diego95's user avatar
  • 511
2 votes
0 answers
142 views

Extension of a given section and obstruction cocyles

Let $p:E \to X$ be a fibration with the fiber $F$ where $X$ is a CW-complex. Denote by $U$ the set $U:=D^n \times \{0\} \cup S^{n-1} \times I$ (part of a cylinder) and let $\tilde{f}:U \to E$ be a ...
truebaran's user avatar
  • 9,200
8 votes
1 answer
641 views

Obstructions for the lifting problem after a pull-back

This is a cross-post from a MSE question which received no answers. Beware that the notation here is a little different. Consider the following lifting problem(s): $\require{AMScd}$ \begin{CD} &...
Cihan's user avatar
  • 1,606
5 votes
1 answer
493 views

Are open orientable 3-manifolds parallelizable via obstruction theory?

In the case of orientable closed $3$-manifolds, we have 4 ingredients that ensure parallelizability: 1a) Closed smooth $n$-manifolds have homotopy type of a $CW$-complex 1b) Closed smooth $n$-...
user6419's user avatar
  • 431
6 votes
0 answers
158 views

Reference to the theorem about linear bundles

The following fact looks like it is well-known. I can prove it myself, but I would like to know of a reference which has a proof? Let $M$ be an $n$-manifold, $C\subset M$ a codimension-1 closed ...
Andrey Ryabichev's user avatar
6 votes
0 answers
115 views

Topological constraints for existing of certain differential operators on manifolds

At the beginning a word of warning: this would be rather vague question: vague as it is, I'm not requiring a precise answer, rather some intuitive explanation. In the flat case $M=\mathbb{R}^n$ ...
truebaran's user avatar
  • 9,200
1 vote
0 answers
124 views

Obstruction theory on $A_{\infty}, C_{\infty}$-algebras

Let $\mathcal{P}_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. Let $A=A^{1}\oplus A^{2}$ be a graded vector space concentrated in degree 1 and 2. Let $m_{n}\: : \:{A^{1}}^{\otimes n}\to A^{2}$ be a ...
Cepu's user avatar
  • 1,424
4 votes
0 answers
188 views

Obstruction to the existence of lifting of the classifying map

Let $E$ be an $n$-plane bundle over CW complex $X$. Then $E$ is a pullback of tautological bundle $\gamma_n$ over $BO(n)$ i.e. $E=f^*(\gamma_n)$. This $f$ is called classyfing map. One can show that ...
truebaran's user avatar
  • 9,200
9 votes
0 answers
392 views

Equivariant obstruction theory done wrong

Let $G$ be a finite group, and let $p:E\to B$ be a $G$-fibration with fibre $F$. The correct framework for studying equivariant sections of $p$ is Bredon cohomology. With the right definitions, most ...
Mark Grant's user avatar
  • 35.4k
7 votes
1 answer
510 views

Topological obstruction for the existence of spin$^c$ structure

Recently I asked on stack exchange the following question: https://math.stackexchange.com/questions/2088888/vanishing-of-certain-cohomology-class-and-existence-of-spin-structure I would like to know ...
truebaran's user avatar
  • 9,200
4 votes
0 answers
178 views

Naturality of primary obstruction under fiber-preserving maps

Let $B$ be a path-connected CW complex, and let $p:E\to B$ and $p':E'\to B$ be fibrations. Let $f:E'\to E$ be a fiber-preserving map, which therefore induces a map of fibers $\bar{f}:F'\to F$. Let us ...
Mark Grant's user avatar
  • 35.4k
4 votes
2 answers
261 views

Asymptotics of list size in Robertson-Seymour theorem

A planar graph cannot have $K_5$ and $K_{3,3}$ as minors. Robertson-Seymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are ...
user avatar
2 votes
0 answers
216 views

obstructions to embeddings of manifolds into Grassmannians

Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in $\...
Shi Q.'s user avatar
  • 543
19 votes
2 answers
1k views

What is obstructing two stably-isomorphic vector bundles from being isomorphic?

The specific situation is the following: Let $n>0$ be a natural number, let $X$ be a finite CW complex of dimension $n$, and let $\xi_0,\xi_1$ be oriented real vector bundles of rank $n$ over $X$ ...
William's user avatar
  • 521
6 votes
2 answers
658 views

Obstructions for $E_n$-algebras

In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure. Have the obstructions for an object ...
user avatar
3 votes
2 answers
555 views

Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...
user109527's user avatar
5 votes
1 answer
599 views

What's a good reference for the following obstruction theory yoga?

Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I ...
Theo Johnson-Freyd's user avatar
0 votes
0 answers
147 views

Extending a 2-frame field - manifolds with boundary

If $M$ is an orientable compact 3-manifold with boundary such that there is defined $s\in\Gamma(\partial M,V_2(TM))$ a section of the 2-frame bundle of $TM$, then $s$ extends to $\tilde s\in\Gamma(M,...
Karthik C's user avatar
  • 261
2 votes
0 answers
114 views

Lazard's $\Gamma_n(f)$ as cocycle

In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...
Jonathan Beardsley's user avatar