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Questions tagged [obstruction-theory]

Obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.

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Obstruction to the splitness of short exact sequence in the category groups

Let $1 \to K \to G \to H \to 1$ be a short exact sequence in the category of groups(interested in non-Abelian groups). My question is the following: Where does the obstruction to the splitting of the ...
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Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
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Vanishing of certain cohomology class and existence of spin structure

Suppose that $M$ is an oriented riemannian manifold and choose transition functions $\varphi_{ij}:U_i \cap U_j \to SO(n)$ for the tangent bundle. They satisfy the cocycle condition $\varphi_{ij} \...
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Obstruction for extending a map along a CW-inclusion: sufficient conditions?

I'm looking for a clarification of the highlighted comment taken from "A User's Guide to Algebraic Topology" by Dodson & Parker. In order to make the setting clear, I uploaded definition $8.1.6$ ...
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Doubts on 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 theory for homotopies

I have a question about obstruction theory extending homotopies. I'm reading Davis & Kirk's chapter 7 (Lecture Notes in Algebraic Topology). They say they consider the problem of "finding a ...
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Lifting problem of a map $g:M\to B^2\mathbb{Z}_n$ to $f:M\to BPSU(n)$

Since there is a short exact sequence of groups: $$1\to\mathbb{Z}_n\to SU(n)\to PSU(n)\to1,$$ we have a fiber sequence: $$B\mathbb{Z}_n\to BSU(n)\to BPSU(n)\stackrel{\iota}{\to} B^2\mathbb{Z}_n\to\...
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Finding explicit form of the first characteristic class of a fiber bundle over $S^2$

These two problems are from an exam I've taken some time ago that I still didn't solve: (1) Does there exist a fiber bundle over $S^2$ with fiber $S^1$, whose characteristic class is equal to ...
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Obstruction to lifting a map from the base space to the total space.

Suppose $\pi :E \to B$ is a fibration with fibre $F$ above a chosen base point. Then I am trying to solve when a map $f$ from a manifold $M$ to $B$ lift to a map $g:M \to E$. The answer given is they ...
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Well-definedness of first obstruction (to extending a map or homotopy of maps over the skeleta of a CW complex) lying in a nonzero group

Let $X,Y$ be CW complexes and suppose for simplicity that $Y$ is simply connected. There is an obstruction $O(f_n)$ to extending a map of CW complexes $f_n:X^n\to Y$ defined only on the $n$-skeleton $...
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Nontrivial obstruction cocycle whose class is trivial

Are there any interesting examples of relative CW complexes $ (X,A) $ where a map $ f \colon X_n \rightarrow Y $ into an abelian space has the property that the obstruction cocycle $ c_f $ is nonzero, ...
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When the primary obstruction is the only obstruction, is it still the only obstruction after a pull-back?

Consider the following situation: We have a Hurewicz fibration $p: E \rightarrow B$ with path-connected base $B$ and $(d-1)$-connected fiber $F$ for some $d \geq 1$. In case $d=1$ we require $\pi_1(...
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Extending map into tangent bundle

Let M be an orientable smooth n-manifold and $p: TM \rightarrow M$ be its tangent bundle. In addition, suppose $f: E \rightarrow M$ is a Hurewicz fibration. Write $e(M) \in H^n(M)$ for the Euler class....
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Showing that the Mobius strip is non-orientable using obstruction theory

In the notes available here, the first example (p. 2) says that the Mobius band is non-orientable because there is an obstruction to a map sending a disk to the band. I don't really understand what's ...
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spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...