# Questions tagged [obstruction-theory]

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

15 questions
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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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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} \... 0answers 169 views ### 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$8.1.6$... 0answers 111 views ### 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 ... 0answers 225 views ### 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 ... 0answers 48 views ### 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\... 0answers 58 views ### 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 ... 0answers 269 views ### 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 ... 0answers 86 views ### 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$...
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, ...