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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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Classification of circle bundles over $\mathbb{RP}^2$

I am trying to understand isomorphism classes of bundles $$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$ These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$. ATTEMPT 1. ...
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Is this inclusion a cofibration?

Let $(X,A)$ be a relative CW-complex. Consider the inclusion $$i:\ \ X\times \{0,1\} \cup A\times I \rightarrow X\times I.$$ I was wondering if this is a cofibration. I guess it is, for there is a ...
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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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Euler class as obstruction to have a never vanishing cross section

We know that (see Hatcher's vector bundles and K-theory Prop. 3.22) the Euler class of an orientable vector bundle or rank $r$, $E\to M$ is the first obstruction to the existence of a never vanishing ...
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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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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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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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Identification of obstructions arising from extending isomorphism between two $U(2)$-bundles over $X$

I need some guidance in see how to use obstruction theory to prove this result. Let $X$ be a closed $4$-manifold, let $E_1,E_2$ be two $U(2)$-bundles over it (I will think of them as vector bundle)...
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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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Examples for obstructions to a $Spin^c$ structure

For a real vector bundle $E$ of rank 3 on the base manifold $M$, the obstruction to having a $Spin^c$ structure on this vector bundle is given by the integral Stiefel-Whitney classes $W_3 = \beta w_2(...
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Almost complex structures on disk bundles

Suppose I have a manifold that is a $n$-disk bundle over $S^n$, i.e a $2n$-dimensional manifold with a single critical point of index $0$ and $n$. I am trying to understand which of these manifolds ...
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Obstruction theory for fibrations.

I'm currently reading Kirk's & Davis's Lecture notes in Algebraic Topology. On the page 190 they discuss obstruction for lifting $f:X \rightarrow B $ to $f':X \rightarrow E$ where $E \rightarrow B$...
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Specific definition of first Stiefel-Whitney class

Let $E \to B$ be real vector bundle. Consider the homomorphism $\pi_1(B) \to \mathbb{Z}_2$ which assign $0$ or $1$ to each loop according to whether orientations of fibers are preserved or not. Since $...
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Mori's “projective manifolds with ample tangent bundles”, Proposition 3

See http://www.jstor.org/stable/1971241?seq=6#page_scan_tab_contents, pages 597 and 598. At the bottom of page 597, we get the following diagram: $\begin{array}{ccccccccc} A/I & \xrightarrow{nat....
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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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1answer
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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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Difference cochain properties

I'm reading about obstruction theory. It's said that difference cochain $\delta (f_n,g_n)$ has properties: $\delta(f_n,g_n)=0$ iff $f_n\simeq g_n (rel X_{n-1})$. $\delta(f,g)-\delta(g,h)=\delta(f,h)....
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Do deformations of a vector bundle form a vector space?

I wonder whether deformations of a vector bundle $F$ over a local Artinian ring $(A, m)$ form a vector space $\operatorname{Ext}^1(F, F) \otimes m$, or what happens in a sequence (*) below? I read ...
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Is there a name given to a space whose first three Stiefel-Whitney classes vanish?

I know that a real manifold $M$ is orientable iff its first Stiefel-Whitney class ($w_1(M)$) vanishes and has Spin structure iff both $w_1(M)$ and $w_2(M)$ vanish. Is there a name given to the ...
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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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1answer
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Extending a section to a basis of sections in a trivial bundle.

Suppose we have a manifold $M$ of dimension $m$ and let $E$ be the rank $n$ trivial vector bundle on $M$. Let $\Gamma$ be a section of the trivial bundle $E$. My question is, are there some conditions ...
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Inversion of Sphere

I was reading about inversion of sphere. Wikipedia defines it as: Let $f: S^2\to R^3$ be the standard embedding; then there is a regular homotopy of immersions $f_t\colon S^2\to R^3$ such that $...
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1answer
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Showing that the Hopf fibration is a non-trivial fibre bundle

I want to show that the Hopf bundle $$ \mathbb{S}^1 \rightarrow \mathbb{S^3} \rightarrow \mathbb{S}^2$$ is non-trivial as a principal fibre bundle. I have seen hints of several different approaches: ...
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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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Trivialised sub-bundle of a trivialised bundle and its orthogonal bundle

I'm reading Kirby's The Topology of $4$ Manifolds, and I encountered the following claim: Recall that a trivialised sub $k$-plane bundle of a trivialised $(n+k)$-plane bundle determines a ...
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Local coefficients involved in the obstruction class for a lift of a map

I'm interested in understanding the importance of the local coefficients in the definition of the obstruction cocycle for a lift of a map $f\colon X \to B$ along a fibration $p \colon E \to B$. I'm ...
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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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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 ...
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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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Vector bundle of dimension $\leqslant n$ on $n$-connected space is trivial

I wonder whether any vector bundle of dimension $\leqslant n$ on an $n$-connected CW-complex is trivial? It seems that, the complex can be given cellular structure with exactly one 0-dimensional cell ...
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Obstruction Theory: extending a map on CW complexes

I'm trying to read about obstruction theory from Davis & Kirk and trying to find a map $ g \colon X_n \rightarrow Y $ from the $ n $-skeleton of a relative CW complex $ (X,A) $ to a path-connected ...
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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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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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1answer
183 views

a question on oriented bundles and Euler class

In Characteristic classes, J. Milnor, J. Stasheff, Prop. 9.7, it is proved that: if the oriented vector bundle $\xi$ possesses a nowhere zero cross section, then the Euler class $e(\xi)=0$. I want ...
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A complex line bundle is trivial if and only if the first Chern class is zero

Let $\xi$ be a complex line bundle over a CW-complex $B$. I want to prove that $\xi$ is trivial if and only if $c_1(\xi)=0$. My attempt: Suppose $c_1(\xi)=0$. Then the Euler class $e(\xi)=0$. Since $...
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2answers
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When does a continuous map $f:X\rightarrow \mathbb{H}P^n$ lift to $S^{4n+3}$?

Consider the Hopf bundles $$S^1\rightarrow S^{2n+1}\rightarrow \mathbb{C}P^n$$ and $$S^3\rightarrow S^{4n+3}\rightarrow \mathbb{H}P^n.$$ In this question (and also here), it is shown that for any ...
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1answer
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Manifolds with non-vanishing vector field and vast homology

Let $n \ge 3$. Is there n-fold $M^n$ with both $\chi(M)=0$ and $\dim H_*(M,\mathbb{R}) \ge$ given number?
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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 ...
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Obstruction to extending $G$-bundle to 4-dimensions in Chern-Simons theory

I am reading Dijkgraaf and Witten's paper on Chern-Simons and finite gauge groups and something they have written about the obstruction to extending the bundle to the 4-manifold confuses me. My ...
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Extending a 2-frame field - manifolds with boundary

It is known that every smooth manifold can be homotoped to a cell complex. In particular this is true for manifolds with boundary. My question: Under the homotopy to a cell complex, is the boundary ...
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If $S^1 \hookrightarrow X$ induces an injection of $H_1$, then $X$ retracts onto $S^1$

This is an exercise in Hatcher, section 4.3, exercise 3, page 419, on which I'm struggling. Suppose that a CW complex X contains a subcomplex $S^1$ such that the inclusion $S^1 \hookrightarrow X$ ...
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A counter example in obstruction theory

Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map $...