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Questions tagged [obstruction-theory]

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4
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0answers
370 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 ...
2
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0answers
58 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 ...
6
votes
1answer
361 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} &...
5
votes
1answer
310 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 $n$-...
6
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0answers
133 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 ...
6
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0answers
109 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 $M=\mathbb{R}^n$ ...
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0answers
94 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 ...
4
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0answers
139 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 ...
8
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0answers
252 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 ...
5
votes
1answer
270 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 ...
4
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0answers
118 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 ...
3
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2answers
188 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 ...
2
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0answers
145 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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2answers
685 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 $X$ ...
6
votes
2answers
488 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 ...
3
votes
2answers
354 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 ...
5
votes
1answer
405 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 ...
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votes
0answers
129 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,...
2
votes
0answers
108 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 ...