Questions tagged [obstruction-theory]
The obstruction-theory tag has no usage guidance.
21
questions with no upvoted or accepted answers
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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 ...
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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 ...
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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 $M=\mathbb{R}^n$ ...
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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 ...
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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 $Q$ ...
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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 ...
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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 ...
- 9,150
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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 ...
- 35k
3
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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 ...
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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 ...
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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
2
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139
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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 ...
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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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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 ...
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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 ...
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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(...
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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, ...
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83
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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 $X,Y,Z$ ...
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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 ...
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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 ...
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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,...
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