# Tagged Questions

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### Collar neighbourhoods for topological manifolds.

The well-known collar neighbourhood theorem states: Let $M$ be a smooth manifold with compact boundary $\partial M$, then there exists a neighbourhood of $\partial M$, which is diffeomorphic to ...
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### Orientation on manifold in terms of homology

One can define the orientation of a manifold $M$ in terms of relative homology groups $H_n (M, M-\{p\})$ by noting that $H_n (M, M-\{p\}) \cong \mathbb Z$ and then designating a generator. Is it ...
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### map of arbitrary degree from compact oriented manifold into sphere

This is a question from a qualifying exam. Let $X$ be a compact, oriented $n$-dimensional manifold. Show that for any $k \in \mathbb{Z}$, there exists a continuous map $f: X \to S^n$ of degree $k$. I ...
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### How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?

I'm in trouble with the following problem: Let $M$ be a manifold with compact boundary $N$ and let $X$ be the double of $M$, that is, the manifold without boundary one gets by glueing $M$ with ...
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### How to Define Product Orientations for Topological Manifolds

When working with smooth manifolds, $M^m$ and $N^n$, it is straightforward to see how orientations at points $p\in M$ and $q\in N$ (i.e. ordered bases for the tangent spaces) give rise to an ...
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### $\mathbb{R}P^2$ and its lines

I have been solving some past exam questions and I came across the following question. Let $r$ and $s$ two distinct lines in $\mathbb{R}P^2$, and let $X$ the space obtained contracting $r \cup s$ to a ...
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### The Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a $3$-manifold which may be viewed as a circle bundle over ...
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### Importance of triangulation

Kervaire's seminal 1960 paper A manifold which does not admit any Differentiable Structure starts "An example of a triangulable closed manifold $M_0$ of dimension 10 will be constructed." What is the ...
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### Is there a compact contractible manifold?

Does there exist a compact connected manifold (without boundary), that has a trivial homotopy type?
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### When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
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### $[M,\mathbb CP^\infty]=[M,\mathbb CP^2]$ where $M$ is a smooth closed orientable 3-manifold!

Prove the above result, where $[X,Y]$ means the set of all homotopy classes of maps from X to Y, two topological spaces. I have answered it below.
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### Fundamental Group, Piecewise Smooth Curves, Consevative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
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### question from hatcher basic 3 manifolds

The question is: why should a homologically trivial embedded sphere in a simply connected (not necessarily compact) 3 manifold M bound a compact 3 manifold embedded in M? I had this problem reading ...
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### Embedded 2-Submanifold

Can you help for solving this problem ı am triying to understand embedded 2 submanifold please help me.
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### Qualifying Exam Question on Manifolds

I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated. Let $P$ be a polygon with an even number of sides. Suppose that the ...