# Questions tagged [geometric-topology]

The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key example is the literature surrounding the Poincaré Conjecture.

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### Prerequisite for Milnor Toplogoy from a Differential Viewpoint

I finished Introduction to Manifold by Loring Tu and I am interested to study this book by Milnor. But I didn't take a course in topology (other than the one I learnt from the appendix of Tu's book). ...
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### Prerequisite for Milnor Topology from a Differential Viewpoint

I finished Introduction to Manifold by Loring Tu and I am interested to study this book by Milnor. But I didn't take a course in topology (other than the one I learnt from the appendix of Tu's book). ...
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### Moduli Spaces of Manifolds vs Classifying Spaces

In the literature, I've seen that people often conflate moduli spaces of manifolds (i.e. a space of submanifolds of a given manifold) with various classifying spaces of diffeomorphism groups. Could ...
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### Concerning the topology of a construction with rectangles

I have a flat space with a rectangular grid put on it, such that the rectangle has width $2r$ and length $h$.The grid has the following property : Take any rectangle $R$. The two rectangles to it's ...
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### Probability on Topological Space

Suppose you have a topological space X, assuming it is Hausdorff, compact, connected space. Is it possible to equip it with probability measure? I am curious if one could create probabilistic ...
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### Is this a valid metric?

I have a model for calculating the distances to SNe Ia (supernovae) which works pretty well on the data. Now, I'm trying to put it in the form of a metric. Here's a time-space diagram describing ...
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### Results about the existence of continuous functions in topology

Given a topological space, what theorems or results are there that shows the existence of continuous function between that space to another space, or between some subset to an open set of the same ...
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### Is there “information theory” for topological space?

Dear stackexchangicians, I have been reading an expository paper about the information theory founded by C. Shannon. The following question is vague, but has been there successful applications of ...
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### Homeomorphism between doughnuts and coffee cups is topology dependent?

please apologize if my language is not appropriate, I am physicist, not mathematician. I am pretty familiarized with the concept of Euler characteristic and the homeomorphism that exist between a ...
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### Conserved quantity along a geodesic in a Killing vector field

Let $(M,g)$ be a Riemannian manifold. Show that if $Y$ is a Killing vector field and $\gamma : (a,b) \rightarrow M$ is a geodesic, then the function $g(\dot{\gamma},Y)$ is conserved along the geodesic....
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### What is homeomorphism? [closed]

In which way I define homeomorphism and how can I explain it with the help of example
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### Why is the Euler class of (the unit tangent bundle over) a manifold equal to its Euler characteristic?

More specifically, I know that the Poincaré-Hopf theorem gives us that $\chi(M)$ is equal to the index of a vector field over $M$. I'm willing to take this as a blackbox, so the question is reduced to ...
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### How to show the inclusion $M\subset\Bbb R^{k+n}$ is locally flat, where $M\subset\Bbb R^k$ is an $n$- dimensional topological manifold?

Source: Madsen & Tornehave, From Calculus to Cohomology Let $M\subset\Bbb R^k$ be an $n$-dimensional topological manifold. Show the inclusion $M\subset\Bbb R^{k+n}$ is locally flat. I.e. ...
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### Project a rectangle onto a sphere

As by the title, I would like to project a finite rectangular object onto a sphere. In particular, as depicted in Img1 there are two different kinds of projections that I would like to perform (A and ...
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### Can a flat Heegaard splitting surface be totally geodesic?

I'm trying to understand (Meeks, Simon, Yau, 1982), page 652, Corollary of Theorem 5: Suppose $N$ is a compact 3-dimensional orientable Riemannian manifold with non-negative Ricci curvature whose ...
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### Critical points and manifolds

This might be trivial but I was wandering if given any manifold $M$ and point $a\in M$ why is it that we can't just define a function that has $a$ as a non-degenerate critical point? I.e., what ...
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### Incompressible surfaces in surface bundles of $S^1$ that are homologous to the fiber are homotopic to the fiber?

I was reading Thurston's "A norm for the homology of 3-manifolds" and I had some questions that I think are pretty basic but have me stumped at the moment. Let $M^3$ be a compact 3-manifold that ...
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### Are a two-petal rose and a Hopf link with joined circles topologically equivalent?

I have a very limited knowledge on topology. From what I can visualize in my mind, a 2-petal rose (or a bouquet of 2 circles, or an 8 figure) in 3 dimensions, is equivalent or homeomorphic to a Hopf ...
### Small exotic $\mathbb{R}^4$'s with symmetries
Definition: An exotic $\mathbb{R}^4$ is a smooth open 4-manifold $E\mathbb{R}^4$ that is homeomorphic to $\mathbb{R}^4$ but not diffeomorphic to it. Definition: An exotic $E\mathbb{R}^4$ is called ...
Consider $\mathbb{R}^4$, and note that $\mathbb{R}^4$ is asymptotically diffeomorphic to $\mathbb{R}\times S^3$, meaning that for some compact subset $K$, $\mathbb{R}^4\setminus K$ is diffeomorphic to ...