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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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Conditions for a lift of an arbitrary map to exist along an arbitrary map

Consider two continuous maps $f: Z \to Y$ and $g: X \to Y$ for (locally finite) CW complexes $X, Y$, and $Z$. I'm interested in knowing conditions, preferably in terms of cohomology, for a lift $h: X \...
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Integral homology isomorphism induces an isomoprhism on weak homotopy classes of maps into a group like space

For context I am working on Weibel's K-book, trying to understand theorem 4.4.3. For this I am reading the paper by Caruso, Cohen, May and Taylor that Weibel refers to. There is a key lemma used in ...
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multiple for first fractional Pontryagin class

Consider the Whitehead tower of the classifying space of the special orthogonal group $BSO(n)$. $$\cdots\to BString(n)\to BSpin(n)\to BSO(n)$$ There is an obstruction to lifting classifying maps into $...
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lie algebra cohomology

Suppose that I have a semi-simple lie algebra $\mathfrak{g}$ and I have a linear map from a vector space $V$ to $\mathfrak{g}$. This map is surjective. Can I show that the corresponding map from $V$ ...
BVquantization's user avatar
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Existence of complex line subbundle of complex bundles

Suppose $M$ be a real smooth manifold, and let $E$ be a rank $r>1$-complex vector bundle over $M$. I would like to know that if there is a complex line subbundle of $E$. By usual obstruction theory,...
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Existence of a global frame of a vector bundle $E$ over $\mathrm{S}^2\times \mathbb{R}^4$

Everything in this question assume the smooth structures. Suppose a vector bundle $E \rightarrow \mathrm{S}^2\times \mathbb{R}^4$ of rank $r$ satisfies the following condition: $E_x := E\big|_{\...
Arith Geo's user avatar
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How to prove that $\mathbb {CP}^\infty$ represents $H^2(-; \mathbb Z)$ **without** using a cell decomposition?

Let $\mathbb P^\infty$ be the infinite-dimensional complex projective space, which is a known $K(\mathbb Z, 2)$, and let $F$ be the functor that assigns to each cell complex $X$ the set of homotopy ...
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relations between obstructions coming from Moore-Postnikov towers

Roughly speaking, I am interested in understanding what (if any) relationships there are between certain obstructions to constructing sections of a bundle of the form $$ P\times_G G/H\to M $$ where $M$...
Sam Ballas's user avatar
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Obstructions to smooth extending the smooth distribution from the boundary.

1. Suppose $(M, \partial M)$ be an $n$-dimensional manifold with boundary, and suppose $E$ be a $(k-1)$-dimensional subbundle of $\mathrm{T} (\partial M)$. Then, could we always find a $k$-dimensional ...
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Obstruction to tensor product decomposition

I'm wondering if anything can be said about the following. Suppose $k \subset K \subset R$ are rings (the case where $k$ and $K$ are fields suffices for me). Is it possible to write $R = K \otimes_k S$...
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Obstruction for a compact simplicial complex

$\mathbf {The \ Problem \ is}:$ Let $X$ be a compact simplicial complex and $Y$ be based, connected space with $f:X\to Y.$ If $X$ is simply connected and $f_k:=f\mid_{X^k}$. Show that $f_1$ is ...
Rabi Kumar Chakraborty's user avatar
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Cohomology class $ H^k(X; \pi_{k-1}(Y,y_0))$ as obstruction

I am following these notes http://scgp.stonybrook.edu/wp-content/uploads/2018/09/lecture-1.pdf on obstruction theory. Let $X$ be a CW complex and denote by $X^{(k)}$ its $k$-skeleton. Suppose that $Y$ ...
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Obstruction Theory in Seifert Surfaces

Background: This is from Livingston unplished notes in knot concordance: Proposition 1.7.1 claims: Let $K$ be a knot in $S^3$. Then every Seifert surface $F$ for $K$ has a function $f:S^3-\nu(K)\to S^...
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The obstruction cochain is zero iff a map has an extension.

On page 142 of this book, we define Definition. Fix $n\geq 1$. A connected space $X$ is said to be $n$-simple if $\pi_1(X)$ is Abelian and acts on the homotopy groups $\pi_q(X)$ for $q\leq n$. ...
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Obstruction theory and maps from a 3-manifold to the sphere

Let $Y$ be a $3$-manifold. Two maps $f,g \colon Y \rightarrow S^2$ are homologous if they are homotopic in the complement of a $3$-ball in $Y$. At the beginning of Section 2.6 of Ozsváth and Szabó's ...
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Classifying the homotopy classes of lifts

Lifts of $f \colon B \to X$ into a fibration $F \to E \to X$ can be identified with sections of the pullback bundle $f^*(E) \to B$. I want to try to compute the path components $\pi_0(\Gamma(f^*(E))$ ...
SourcedDirect's user avatar
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About the obstruction theory and homotopy?

I am trying to read Hatcher's vector bundle and $K$-theory. I am trying to understand the obstruction theory in Chapter $3$. When we get a vector bundle $\pi:E\rightarrow B$ and we may suppose that $B$...
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Why are principal K(A,n) fibrations induced by pullback

I am trying to classify such fibrations $p:E\to B$. I understand how to build a cochain representing the obstruction to building a section, and showing it is a cocycle, getting a cohomology class in $...
Ryan's user avatar
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Can I lift the maps from $S^2 \times S^3 \rightarrow \frac{U(n)}{O(n)}$ to $S^2 \times S^3 \rightarrow U(n)$

For $n\geq 3$, I am considering all maps $S^2 \times S^3 \xrightarrow{f} \frac{U(n)}{O(n)}$ I wish to know whether there exists a lift to $S^2 \times S^3 \xrightarrow{\tilde{f}} U(n)$ where the map $U(...
Yen-Ta Huang's user avatar
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Computing a Gromov-Witten invariant

Some background that is not necessary for answering the question: Let $X = \mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ be a threefold. This is a $\mathbb{P}^1$-bundle ...
Nachiketa's user avatar
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Cohomology with local coefficients and obstruction theory

I'm reading about obstruction theory on Milnor & Stasheff and came across the following claim: If $p:E(\xi)\rightarrow B$ is a vector bundle over a CW complex $B$ and $V_k(\xi)$ is the ...
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Topological obstruction to time-reversal symmetry

Let $M$ be a (compact if this somehow matters) smooth manifold with a involution $\theta: M \to M$, i.e. a smooth map such that $\theta^2 = \text{Id}$. A typical example is $M$ a $d$-torus and $\theta(...
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Basic obstruction theory : where does the obstruction to uniqueness of lifting lie?

This is a question about a remark someone said to me without giving much precision. Suppose you have two nice spaces $X,Y$ and are trying to build a map $X\to Y$ with certain nice properties. ...
Maxime Ramzi's user avatar
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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. $...
Overflowian's user avatar
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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 ...
Aolong Li's user avatar
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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\...
Borromean's user avatar
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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....
Cihan's user avatar
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4 votes
1 answer
124 views

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)...
Riccardo's user avatar
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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 a) ...
Maksim Dolgikh's user avatar
2 votes
1 answer
266 views

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(...
Chao-Ming Jian's user avatar
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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 ...
user39598's user avatar
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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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1 answer
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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....
numberjedi's user avatar
6 votes
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245 views

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} \...
truebaran's user avatar
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1 vote
1 answer
381 views

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 ...
Math's user avatar
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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)....
alans's user avatar
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2 votes
1 answer
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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 ...
evgeny's user avatar
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8 votes
1 answer
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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 ...
R_D's user avatar
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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 ...
happymath's user avatar
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3 votes
1 answer
295 views

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 ...
happymath's user avatar
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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 $...
happymath's user avatar
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6 votes
1 answer
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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: ...
Timon van der Berg's user avatar
1 vote
0 answers
121 views

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 ...
Dan Armour's user avatar
4 votes
1 answer
166 views

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 ...
Luigi M's user avatar
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4 votes
1 answer
804 views

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 ...
Riccardo's user avatar
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