Tagged Questions

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Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
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Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
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Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
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Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
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Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $R^2-\{0 \}$ is zero, ...
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Turning a torus inside-out

Smale's paradox is now famous, and great videos can be found illustrating it. Similarly, there is a video showing how to turn a torus inside-out. The solution seems to be simpler, but is the proof ...
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How to kill homotopy groups using framed cobordism

Let $M$ be an orientable manifold (with or without boundary), $N$ a framed submanifold in the interior of $M$ and assume (if necessary) that $\dim N<(\dim M)/2$. If some low-dimensional homotopy ...
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Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
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Euler class is odd under orientation, thus its integral over a manifold will be even.

I learned a statement from others: "Euler class is odd under orientation, thus its integral over a manifold $M$ is even." I cannot fully appreciate it, can someone show this explicitly? The ...
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Avoiding self-intersections of immersed manifolds

Let $i: N\to M$ be an immersion of manifolds. If $\dim M\geq 2\dim N+1$ (or something like that?), does there exist arbitrary small perturbations of $i$ (wrt. some reasonable norm) that are already ...
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invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
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Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y)$ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$. ...
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Another differential topology lemma

Another lemma (1) Why can we assume $z=f(z)=0$ and that $U$ is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ...
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Two books defined two Chern(Euler) classes yet differed by a negative sign, what's wrong?

In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data ...
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A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
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Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
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Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
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Reviewing the basics of algebraic topology for further deeper study

I am entering a Ph.D. program in pure math this fall. Over the summer, I am hoping to review the basics of algebraic topology (fundamental group, covering space theory, etc.) and I wondered if you had ...
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express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space ...
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let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
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tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
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History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
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Tangent bundle of a manifold [duplicate]

can anyone help me with this problem: Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$? for the 1st ...
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Construction of Hodge decomposition

We know Hodge decomposition splits any $k$-form into three $L^2$ components. And I see some proofs, none of them provide an explicit constructive method. Is there any general method to construct one? ...
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English edition of Vol 9 of Dieudonné's Foundations of Modern Analysis?

I have found the first 8 volumes of DieudonnÃ©'s Foundations of Modern Analysis in English translation, but I'm having difficulty locating volume 9. I have searched the catalogues of numerous libraries ...
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vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
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Reference about history of characteristic classes

I'm looking for a good reference about history of characteristic classes. For example, I would like to know how Chern define the Chern class at first in history, or who rewrote the characteristic ...
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References Request (Poincare Duality via Unimodular pairing, de Rham isomorphism)

Chapter 0 of Griffith's Harris contains a substantial amount of topological results that I have not seen elsewhere, namely: Poincare duality as a unimodular intersection pairing on homology. Also ...
I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...