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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25 views

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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Hodge decomposition in weight Sobolev space

Let $(M, g)$ be a Riemannian manifold with cylindrical ends, i.e., the ends of $(M, g)$ are modelled on $([0, \infty)_s\times Y, g=ds^2 +g_{Y}).$ I want to know whether the Hodge decomposition for $L^...
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Continuously summing a family of maps.

Let $X$ be a space and $$\{f_i:X\rightarrow[0,\infty)\}_{i\in\mathcal{I}}$$ a family of continuous maps $X\rightarrow [0,\infty)$ indexed by some set $\mathcal{I}$. Assume that the family is point-...
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Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$

I found one rather interesting but intractable topology problem. Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$ Despite various attempts to contract ...
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Does this count as a loop on Möbius strip

Suppose we take the Möbius strip as $X = \frac{[0, 1]\times[0, 1]}{\sim}$ with usual equivalence relation. If I define $\alpha: [0, 1] \rightarrow X$ by $x \rightarrow [(x, 1/2)]$, is this a loop? ...
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Quotient of two classifying spaces? [closed]

Suppose both $G$ and $K$ are continuous Lie groups, $G/K$ is a homogeneous space (also a quotient space) but not a group. Is it true that the quotient of the classifying spaces $$(BG)/(BK)$$ is a ...
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Connection of product spaces with constraints [closed]

I have a question. If I have the following topological space $\mathbb{R}^2\times\mathbb{S}^1\times\mathbb{S}^1$ that corresponds to the following coordinate vector in $\mathbb{R}^4$, $x= [x_1,x_2,x_3,...
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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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Euler Characteristic for relative cell-complex, show that $\chi(A)-\chi(X)+\chi(X,A)=0$.

Let $X$ be a finite cell-complex and $A\subset X$ be a sub-cell complex. The post has been answered here: Euler characteristic for CW complexes asked about the proof for $$\chi(A)-\chi(X)+\chi(X/A)=...
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Why is the attaching map for the top cell in the connected sum denoted by “sum” of attaching maps of top cells in the manifolds?

In a book I read: $M$ and $N$ are $d$-dimensional manifolds. Let $\widetilde{M}$ be the $(d−1)$-skeleton of $M$, or equivalently, $\widetilde{M}$ is obtained from $M$ by removing a disc in the ...
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What is a representation of a (branched) cover?

I'm reading a paper about finite-order homeomorphisms of a closed oriented surface, say $f: M \rightarrow M$, $f^n = id$. I know that from such an $f$ we get a covering $P: M \rightarrow M_f$, where $...
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Does an analogue of the sphere theorem hold in higher dimensions?

The sphere theorem of Papakyriakopoulos states that if $X$ is a 3-manifold with non-trivial $\pi_2(X)$, then some non-zero element of $\pi_2(X)$ is represented by an embedding $\mathbb S^2 \to M$ (...
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Prove the set of continuous functions on $S^1$ is isomorphic to the set of continuous periodic functions with period $2\pi$

Let $S^1 = \{ (x,y)\in \mathbb{R}^2 | x^2+y^2=1$. Let $z:\mathbb{R} \to S^1$ given by $z=(cos(\theta),sin(\theta))$. Define the map $z^* : C^0 (S^1) \to C^0 (\mathbb{R})$ by $z^* (f)= f \circ z$. I'm ...
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How to shrink 2D complex shapes in themselves?

Is there a robust method that can shrink 2D shapes in themselves like seen on the images? I am talking about more complex shapes than the ones on the image. Shapes that don't necessarily have such a ...
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Is there a name for such manifold? Generalization of Grassmanian

Let $V$ be an $n$-dimensional vector space over a field $K$. The Grassmannian $Gr(k, V)$ is the set of all $k$-dimensional linear subspaces of $V$. Suppose a manifold is defined as $$ G' (k, V) = ...
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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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Distortion of the Unknot

In Mikhail Gromov's "Filling Riemannian Manifolds" he defines the distortion of a knot $K$ embedded in $S^3$ as $$\delta (K) := \inf_{\gamma \in K} \sup_{x,y \in \gamma} \frac{d_{\gamma}(x,y)}{||x-y||}...
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How to determine homotopy equivalence of wedge sum?

I am very stuck on the following problem despite my repeated attempt to prove it. The problem is from Topology II book of Encyclopaedia of Mathematical Sciences from Springer: Let M and N be ...
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Euler Characteristic of CW Complex

The following problem comes from Lundell’s book on CW complex. I am stuck with it: Let $p : E \rightarrow B$ be a fiber bundle with fiber $F$. Suppose that $F$ is a finite CW-complex and $B$ is a ...
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How to determine the Cohomology of Wedge Sum of Manifolds?

Since few days, I have been struggling to prove the following statement from Topology II and Lundell’s book: Let us assume M and N are positive dimensional closed manifolds. Prove that the wedge sum ...
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1answer
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Seifert Fibered Spaces with Boundary are $\mathbb{P}^2$-irreducible

I'm reading Peter Scott's The Geometry of 3-Manifolds and am trying to understand the argument behind this statement, which arises in the proof of Corollary 3.3: If $M$ is a Seifert fibered 3-...
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Identifying $UT\mathbb{H}^2$ with $PSL(2,\mathbb{R})$

I'm trying understand more fully this sentence in Peter Scott's "The Geometries of 3-manifolds" he says: "$PSL(2,\mathbb{R})$ acts transitively on $UT\mathbb{H}^2$ and the stabilizer of a point of $...
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Cohomology Groups of Projective Space and Sphere

I am stuck on the following question from Switzer’s Algebraic Topology: Homotopy & Homology: Describe the cohomology groups of $\mathbb{R}P^2 \times \mathbb{R}P^2$ and $\mathbb{S}^m \times \...
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Characterizing the Homotopy of Path-Connected Space [duplicate]

I am currently really stuck and confused about the following problem from Topology II of Encyclopaedia of Mathematical Sciences (Springer, Novikov/Fuchs): Give an example of two path-connected spaces ...
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Use space suspension and reduced homology to compute the homology group $H_{k}(\mathbb{S}^{n})$ for any $n\geq 0$.

For a space $X$, the suspension of it is defined by $$\Sigma X:=C_{+}(X)\cup_{X} C_{-}(X),$$ where $C_{+}(X)$ and $C_{-}(X)$ are the upper and lower cone of it. The reduced homology, roughly speaking,...
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Confusion on the kernel and image of boundary map in terms of computing simplicial homology.

I am learning simplicial homology, and I have a confusion on the computation. I understand that the boundary map the boundary map $\partial_{n}:C_{n}(X)\longrightarrow C_{n-1}(X)$ takes the form $$\...
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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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1answer
108 views

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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1answer
31 views

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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1answer
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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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Does the image of the cartesian product of mobius strips under a map intersects itself?

Context: I'm thinking about using mobius strip to represent pairs of points on a simple close curve (on $\Bbb{R}^2$ ,that doesn't intersect itself) which can be constructed by $\gamma:[0,1]\to\Bbb{R}^...
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33 views

Show that a Quotient Map pasting a polygonal region is closed

Given a polygonal region $P$, a labeling equivalence relation and a pasting quotient map $\pi: P \to X$, show that $\pi$ is closed. For any set $C$ closed in $P$, show that $\pi(C)$ is closed. I'm ...
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Can we extend a function on surface to a function on $\mathbb{R}^3$

I'm studying differential geometry right now. I've encountered a problem considering the definition of a function/differential forms on a surface $S \in \mathbb{R}^3$. Suppose we're given a $C^1$ ...
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1answer
37 views

Reference request for surgery on knots

I've seen this article https://www.math.cuhk.edu.hk/~ztwu/JonesCosmetic.pdf on the Jones Polynomial and Cosmetic Surgery and I've looked at the Wikipedia entry on Dehn surgery as well. My background ...
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1answer
37 views

Existence of criteria for extension of action of embedded circles in a topological d-manifold.

Let $M$ be a topological d-manifold viewed as an embedded subspace $M \hookrightarrow \mathbb{R}^{m}$ for some $m$. Let $End(\mathbb{S^1}) = Homeo(\mathbb{S^1})$ be the group of homeomorphisms of the ...
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49 views

Does the figure eight ($S^1\vee S^1$) have the fixed point property? [duplicate]

Does the figure eight (The wedge sum of two circle $\simeq S^1\vee S^1$) has the fixed point property or not? How would you justify your answer?
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1answer
39 views

Determine the cohomology groups of complex projective space

S.E. Mathematicians, I have trouble solving the following problem: Let $M = \mathbb{C}P^3$ with a point removed. Determine the cohomology group $H^i (M,\mathbb{Z})$ and compact support cohomology $...
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1answer
32 views

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 ...
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42 views

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 ...
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Uniqueness of asymptotic topology?

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 ...

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