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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 ...

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A difficulty in understanding a part of solution of Q.1.3.10 in Allan Pollack and Guillemin.

"Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold Z of X. Suppose that for all $x \in Z$, $$df_{x}: T_{x}(X) \rightarrow T_{f(...
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A difficulty in understanding a part of the solution of Q.10 section 1.3 in Allan Pollack and Guillemin(3).

Q.10 section 1.3 in Allan Pollack and Guillemin is the following: "Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold Z of X. ...
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First generalization of the inverse function theorem Q.10 section 1.3 in Allan Pollack and Guillemin(2).

Part of Q.10 section 1.3 in Allan Pollack and Guillemin is the following: "Generalization of the inverse function theorem.let $f:X \rightarrow Y$ be a smooth map that is 1-1 on a compact submanifold ...
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Nowhere vanishing vector field on Moebius strip

I know there is no continuos non-vanishing normal vector field on Moebius strip, which is pretty obvious. Is it possible to construct a nowhere vanishing tangent vector field on Moebius strip? ...
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Geometric interpretation of torsion homology classes

Suppose I have a homology class $x \in H_1(M)$ which is torsion of order $k$ say. Suppose furthermore that $M$ has Dimension big enough, such that every element of $H_1$ and $H_2$ can be relalized as ...
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One-degree map between manifolds with boundary

Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[...
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To what extent are homeomorphisms just deformations?

Background. It is often said that two spaces are homeomorphic if, roughly speaking, one space can be continuously deformed into the other without any tearing and gluing. It is then emphasized that ...
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$\mathbb{HP}^2$, exotic 7-spheres, and Bott manifolds

I am looking for some explanation how $\mathbb{HP}^2$, exotic 7-spheres, and Bott manifolds are related? And how the construction of a Bott manifold is related to $\mathbb{HP}^2$ ...
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Linking number and cup product

Let $S^p$ and $S^q$ be disjoint spheres in $\mathbb{R}^n$ with $n=p+q+1$ and let $X= \mathbb{R}^n- (S^p\cup S^q)$. By Alexander duality, their fundamental classes yield cohomology classes in $\tilde{H}...
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The Problem visualizing a 3-dimensional surface . [closed]

We know from the generalized volume formula of the sphere for any dimensions , the surface area of a 3-sphere is twice pi squared times r cubed , where r=the radius . Of course the surface is 3-...
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Deform surface along some flow

I am designing a software in which the user can cut a surface (such as a sphere or a torus) along some (closed) curve. I would then like the surface to 'unfold' in some way, for example cutting a ...
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$\Omega_4^{SO}(K(\mathbb{Z}_2,2))$ v.s. $H^4(K(\mathbb{Z}_2,2),U(1))$: Cocycle form

The $SO$ bordism group of Eilenberg–MacLane space $K(\mathbb{Z}_2,2)$ is $\Omega_4^{SO}(K(\mathbb{Z}_2,2))=\mathbb{Z}_4$. The cohomology group of $K(\mathbb{Z}_2,2)$ with $\mathbb{R}/\mathbb{Z}=U(1)$...
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Covering space in B. O'Neill book, semi-riemannian geom.

i'm reading the Appendix A in B. O'Neill's book semi-riemannian geometry and he says this. I'm thinking on $\text{exp}:\mathbb{R}\to S^1$ dose not provide an izo at the level of fundamental groups. ...
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Is it there a nice way of visualising $\pi_3(\mathbb{S}^2)$?

I was wandering if there is a nice geometric way of visualising $\pi_3(\mathbb{S}^2)$. Thanks in advance.
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Pontryagin square: lifting to an integral cocycle, and an element $H^4 (B^2 \mathbb{Z}_r, \mathbb{Z}_{2r})=$Hom$(\mathbb{Z}_{2r},\mathbb{Z}_{2r})$?

Let $M$ be a simplicial complex and $Π$ be a finite abelian group. In the simplest case $Π = \mathbb{Z}_r$ finite group with $r$ even, the Pontryagin square is a cohomological operation which maps an ...
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Sum formulas for Pontrjagin square and Postnikov square

Inspire by this, I wonder Pontrjagin square: There is a geometric interpretation of $\mathfrak{P}_2$, due to Morita. Assume $q=2k$, so that the Pontrjagin square is a map $$\mathfrak{P}_2 \colon H^{...
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Kirby–Siebenmann class and the 4th Stiefel-Whitney class: $ \operatorname {ks} (M)$ v.s. $w_4(M)$

Kirby–Siebenmann class $ \operatorname {ks} (M)$ is an element of the fourth cohomology group $$ {\displaystyle \operatorname {ks} (M)\in H^{4}(M;\mathbb {Z} /2)} $$ which must vanish if a ...
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Cobordism theory for piecewise-linear (PL) and topological manifolds

The Cobordism theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise-linear and topological manifolds. I know the ...
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An intrinsically curved surface with no extrinsic curvature

An intrinsic curvature is the property of a manifold itself. It shows it's deviation from euclidean geometry. The extrinsic curvature, on the other hand, depends on how the surface is embedded in a ...
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$E_8 \oplus m[1]$ ever diagonal?

Let $E_8$ be the unique unimodular positive definite even integral symmetric form of rank 8. Let $[1]$ denote the unique unimodular positive definite even integral symmetric form of rank 1. Is $E_8 \...
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Cobordant of Dold manifold and Wu manifold via fibered classifying spaces

Background: I think, Dold manifold and Wu manifold are 5-dimensional manifolds which are cobordant to each other via 5-dimensional bordism group: $$ \Omega^{SO}_5. $$ Literally, cobordism theories ...
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One-degree map between orientable compact manifolds

Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[...
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Poincaré dual of the generators of $H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)$

We know $H^d(S^5/\mathbb{Z}_4,\mathbb{Z}_4)=\mathbb{Z}_4$. So there are two classes of $\mathbb{Z}_4$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$. Wha are the Poincaré dual $(5-d)$-...
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Poincaré dual of the trivial class $H^1(\mathbb{RP}^5,\mathbb{Z}_2)$

Let $$a \in H^1(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2,$$ When $a' \in H^1(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2$ is a nontrivial generator, the Poincaré dual (4-manifold generator) PD$(a')$ of ...
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Is there a rule for “kinks” in boundaries of point sets?

Consider sets $A$, $B$, $C$ each being a subset of $\mathbb{R}^2$ and a set $D := (A \cap B) \cup C$. For the given examples the following obervation can be made: each of those marked red points $p_{R}...
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Poincaré dual of $H^1(M,\mathbb{Z}_2)$ for a $\frac{\mathbb{CP}^2\times S^1}{\tau}$

Given a $M=\frac{\mathbb{CP}^2\times S^1}{\tau}$, where $τ$ acts as $−1$ on the sphere $S^1$ and a complex conjugation on complex projective space $\mathbb{CP}^2$. See Dold, Albrecht (1956), "...
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Submanifold generator and its Poincaré dual in a 5-dimensional real projective space

1) What is the nontrivial 1-dimensional sub-manifold generator $M^1$ in $\mathbb{RP}^5$ of $H^1(\mathbb{RP}^5,\mathbb{Z}_2)$? How to visualize it where this $M^1$ sits in $\mathbb{RP}^5$? 2) What ...
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3D lattice of a tetrahedron. What is it called?

I recently stumbled on this image and have been looking for a name for it: pyramid image http://tetraktys.de/bilder/buckminster-tetraeder.gif It’s not a Seirpinski pyramid because it doesn’t become ...
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The Distance between two points in a hypothetical universe.

I have a hypothetical universe where the distance between two points in spacetime is defined as:$$ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2+dz^2$$Where $\phi$ has units of $km\space s^{-2}$. The space in ...
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Does this manifold have a name?

While using Mathematica to alter manifolds and numerically verify the Gauss-Bonnet Theorem, I generated the figure whose pictures I've attached in this post (multiple perspectives of the same shape ...
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Submanifold generators that correspond to Stiefel Whitney class of $\mathbb{RP}^n$

Here let me write the real projective space $\mathbb{RP}^n \equiv RP^n$. I computed that $$w_1(RP^5)=0,$$ $$w_2(RP^5) \in H^2(RP^5, Z_2) \neq 0,$$ thus it is non-zero. $$w_3(RP^5)=0,$$ $$...
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Justifying casual claims in algebraic topology [closed]

Especially when you study algebraic topology, you'll encounter a lot of casually stated claims, involving homeomorphisms, quotient space constructions, or deformation retracts. One very famous example ...
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notation for connected sum $\#^n S^2 \times S^2$

What does the symbol $\#^n S^2 \times S^2$ mean in geometric topology? I know the $\#$ symbol refers to a connected sum. So that we delete a disk from each sphere and sew the two spheres according ...
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Does the Poincare homology sphere smoothly embed in $\mathbb{C}P^2$?

Does the Poincare homology sphere smoothly embed in $\mathbb{C}P^2$? If so I guess one way to see this would be to have a handle decomposition for $\mathbb{C}P^2$ such that a subset of the 2 handles, ...
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How to visualize the String(n) group?

I am trying to get some more intuitions about the statement: Killing the $\pi_3$ homotopy group in Spin(n), one obtains the infinite-dimensional string group String(n). Formally: I know that here ...
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Induced stable framing on circles

Let $C$ be an embedded circle in $\mathbb R^n$. Then any trivialization induces a stable framing of the normal bundle $\nu(C)$ of $C$ in $\mathbb R^n$ in the following way: $$ TL \oplus \nu(C) \cong ...
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Understanding Adams spectral sequence and Pontryagin-Thom isomorphism intuitively

The question is about understanding Adams spectral sequence intuitively and some of the meanings of its relations. In Adams spectral sequence, $$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTG), \...
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Understanding De/Suspension $\Sigma^{-1}(\Sigma{X})\neq X$

The question is about understanding suspension and desuspension, see also a previous question. Question: How do we define desuspension exactly? (Please see the comments below, people complain about ...
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Can two ellipsoids meet in a pair of ellipses intersecting in four points?

On the plane two ellipses can intersect in exactly four different points. In space two ellipsoids can meet in a pair of ellipses intersecting in exactly two different points. For example take $x^{2}+...
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The suspension (topology) and elementary examples

Let $\Sigma$ denotes a suspension $$\Sigma X =S^1 \wedge X \equiv (S^1\times X)/(X\vee S^1)$$ where $\wedge$ is the the smash product, and $\vee$ is the wedge sum (one point union) of pointed ...
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The pattern of Mills Mess juggling and a link invariant in 3D?

You can find the juggling pattern of the 3 balls called the Mills Mess juggling here. And a [simplified animation is found here] Observation and Attempt: - Notice that for each of the three balls, ...
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Understand the suspension (topology) and some Lie group examples

Let $\Sigma$ denotes a suspension $$\Sigma X =S^1 \wedge X \equiv (S^1\times X)/(X\vee S^1)$$ where $\wedge$ and $\vee$ are the smash product and the wedge sum (one point union) of pointed ...
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Atiyah-(Patodi)-Singer index theorem and an instanton on $S^5$

In this research paper, it states a math-physical description of the particular instance of Atiyah-(Patodi)-Singer index theorem. In a footnote 2, it says: If one conformally compactifies $S^4 \...
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$\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$?

From the computation of some lower dimension $N$ of $Sp(N)$ group, we see that the homotopy groups are: $\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$, at ...
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Terminology for topologies that can't be embedded in a plane (without severing the topology)?

In some code I've written, I justify the existence of a certain structure as follows: X is needed because we support topologies that cannot result in a planar embedding, e.g. cube, sphere, torus, ...
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How to pair the Arf with Stiefel-Whitney class?

The Arf invariant is a nonsingular quadratic form over a field of characteristic 2. The form that I looked at was: $$ S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...
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Valuation Property for mean width

For some polyhedron, $P$, define the mean width function, $$H(P)=\sum_{e\in E} L_e(\pi - \delta_e)/(4\pi)$$ Where $E$ is the set of all edges of the polyhedron, $L_e$ is the length of edge $e$ and $\...
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From $\mathbb{R}^4$ to a topological sphere $S^2$

In Wikipedia it says: If spacetime is $\mathbb{R}^4$, the space of all possible connections of the $G$-bundle is connected. But consider what happens when we remove a timelike worldline from ...
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General Stiefel-Whitney classes and Stiefel manifolds

Here are some statements that I wish to understand more deeply, whose truth value I want to check, and to determine under which criteria they are valid. Consider the Stiefel manifold $V_k(R^n)$ of ...
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Stiefel–Whitney class, obstructions and exact sequences

The first Stiefel–Whitney class $w_1$ is zero if and only if the bundle is orientable. In particular, a manifold $M$ is orientable if and only if $w_1(TM) = 0$. The first and second Stiefel–Whitney ...