Questions tagged [obstruction-theory]
The obstruction-theory tag has no summary.
56 questions
8 votes
0 answers
307 views
Subtle gap between PL & SMOOTH in dimension 4
Main Question and Subtlety The folk wisdom, often cited as a clear truth, states, "PL = SMOOTH in dimension 4". While intuitively appealing, this is not a precise mathematical statement. The ...
- 5,748
6 votes
2 answers
418 views
What obstructions does the failure to agree between $H^3(\mathfrak g, \mathbb K)$ and $\text{Sym}^2(\mathfrak g)^\mathfrak g$ measure?
It is known that for semisimple Lie algebras, one has an isomorphism: $$H^3(\mathfrak g, \mathbb K)\cong\text{Sym}^2(\mathfrak g)^\mathfrak g$$ Where the latter are the invariant symmetric bilinear ...
- 531
4 votes
1 answer
227 views
Moduli of stable maps for symplectic varieties
Consider holomorphic symplectic manifold $T^*\mathbb{P}^1$ and moduli of degree $n$ stable map with $3$ marked points $\mathcal{M}_{0,3}(T^*\mathbb{P}^1, n)$. Then the virtual dimension of $\mathcal{M}...
- 563
4 votes
1 answer
316 views
Obstruction for a BoP orientation?
Pengelley defined the BoP spectrum and showed its similarity to ko. A ko orientation corresponds to a spin structure. Is this also true for BoP? I.e. what is the obstruction for a BoP orientation?
- 263
14 votes
1 answer
1k views
Higher Obstruction Theory?
Obstruction theory is an intuitive yet powerful method in homotopy theory. It helps study the space of maps (more generally, sections of bundles) inductively on skeletons. For example, it provides a ...
- 5,748
9 votes
0 answers
169 views
Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$ are homotopy equivalent. Moreover,...
- 381
4 votes
0 answers
282 views
Possible obstructions to global Wick-rotation in distinguishing spacetimes
Take the time-orientable $3+1$ dimensional spacetime $(M,g)$ that is locally Wick-rotatable at any arbitrary point $p \in M$ to a Riemannian manifold $(N,h)$. Local Wick-rotatability of $(M,g)$ ...
- 658
0 votes
0 answers
244 views
Are causally isomorphic spacetimes Wick-related?
Take the time-orientable spacetimes $(M_1,g_1)$ and $(M_2,g_2)$ that are locally(to be clarified below) Wick-related and both are globally Wick-rotatable(to be clarified below) to the same Riemannian ...
- 658
2 votes
0 answers
148 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 ...
- 51
4 votes
0 answers
257 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 $Q$ ...
0 votes
1 answer
225 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 ...
- 3,432
15 votes
2 answers
2k 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 ...
- 658
2 votes
0 answers
396 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 ...
- 658
4 votes
1 answer
182 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 ...
- 1,273
2 votes
0 answers
150 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 ...
- 213
2 votes
1 answer
214 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 $$ 0 \to H^k(...
- 303
9 votes
1 answer
381 views
Non-triviality of a Postnikov class in $H^3\left(B \operatorname{PSU}(N) ; \mathbb{Z}_q\right)$
Let $\alpha\in H^2(B\operatorname{PSU}(N) ; \mathbb{Z}_N)$ be the obstruction class for lifting a $\operatorname{PSU}(N)$-bundle to an $\mathrm{SU}(N)$-bundle. Note that $\operatorname{PSU}(N)\cong \...
- 2,373
0 votes
0 answers
96 views
Obstruction to finding a framing for quotient manifolds
The question is rather open-ended but I hope it is concrete enough. If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what ...
7 votes
2 answers
700 views
Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence $$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$ where $G_c$ is the normal subgroup which ...
- 125
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< ...
- 2,562
8 votes
1 answer
559 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 ...
- 20.2k
10 votes
2 answers
500 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 ...
- 2,350
3 votes
0 answers
232 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 ...
- 111
1 vote
0 answers
223 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(...
- 1,536
9 votes
1 answer
355 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, ...
- 28.7k
1 vote
0 answers
93 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, ...
- 915
1 vote
0 answers
104 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 $X,Y,Z$ ...
- 599
4 votes
1 answer
161 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 ...
- 243
3 votes
1 answer
275 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 ...
- 1,228
5 votes
0 answers
345 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 ...
- 66.6k
2 votes
0 answers
230 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 \...
- 131
3 votes
1 answer
454 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 ...
- 2,562
2 votes
2 answers
256 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
862 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 ...
- 5,748
8 votes
1 answer
718 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,...
- 2,562
12 votes
1 answer
983 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 ...
- 66.6k
12 votes
1 answer
2k 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 $...
- 2,562
5 votes
0 answers
822 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 ...
- 521
2 votes
0 answers
204 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 ...
- 9,634
8 votes
1 answer
725 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} &...
- 1,936
7 votes
1 answer
566 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 $n$-...
- 471
6 votes
0 answers
181 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 ...
- 1,095
6 votes
0 answers
147 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 $M=\mathbb{R}^n$ ...
- 9,634
1 vote
0 answers
144 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 ...
- 1,434
4 votes
0 answers
225 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 ...
- 9,634
9 votes
0 answers
492 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 ...
- 37.2k
7 votes
1 answer
620 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 ...
- 9,634
4 votes
0 answers
206 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 ...
- 37.2k
4 votes
2 answers
316 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 ...
user76479
2 votes
0 answers
255 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 $\...
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