Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
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Categorical Notion of Quotient in Spectra

Having done some reading on spectra recently, I noticed that the definition of a quotient spectra for a closed subspectrum of a CW spectrum is simply given by taking the quotient of each of the spaces ...
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what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
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Fundamental group of $\mathbb{S}^{n-1}\times\mathbb{R}$ minus $k$ disks $\mathbb{D}^n$

Let $X$ be the space obtained from $\mathbb{S}^{n-1}\times\mathbb{R}$ by deleting $k$ disjoint subsets, each one homeomorphic to $D^n$. What is the foundamental group of $X$?
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Graph embedding into a surface

For example, let's consider a $K_{5}$ (complete graph on 5 vertixes) and a torus, which is defined as $S^{1} \times S^{1}$. How to build a continous embedding $f:K_{5} \rightarrow \mathbb{T}^{2}$? We ...
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If $G$ is a finite nontrivial group, then $K(G, 1)$ cannot be a finite CW-complex?

I am currently working on this algebraic topology problem and got stuck: Suppose $X$ is a finite CW complex with $\pi_1(X)$ a nontrivial finite group. Show that its universal cover $\widetilde{X}$ ...
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Does every continuous action of $S^1$ on $R^n$ have a fixed point?

I certainly can't think of one that doesn't. I am aware that there are decompositions of $R^n$ as a union of embedded $S^1$'s, but none of these seem like they would support a continuous action. ...
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Find a presentation for the fundamental group of $P^2\#T$

I have to find a presentation for the fundamental group of $P^2\# T$. Which I believe to be identified by the following labeling scheme: $(a_1b_1a_1b_1)(a_2b_2a_2^{-1}b_2^{-1})$. $P^2$ is the ...
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Weak equivalence of ordinary and homotopy colimits

I am looking for conditions under which colimits and homotopy colimits of diagrams of, say, topological spaces, are weakly equivalent. I would appreciate answers not demanding an all too profound ...
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canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization ...
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Ideas for basic application of homotopy theory to homological algebra?

I'm taking a first course in homological algebra. As a project, the lecturer suggested each student find a topic, presentable in an hour, relating to the material studied in the course. The material ...
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Is there a compact manifold having Euler characteristic 0 which cannot be given a Lie group structure?

I realized that a (compact) Lie group must have Euler characteristic 0 due to Poincare-Hopf index theorem. Now I'm thinking of its converse. Is there a compact manifold having Euler characteristic 0 ...
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underlying real vector bundle of a complex vector bundle

Let $\eta^\mathbb{C}$ be a complex line bundle. If the underlying $2$-dimensional vector bundle $\eta$ is not trivial as a real vector bundle, can we obtain that $\eta^\mathbb{C}$ is not trivial as a ...
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Homology of a co-h-space manifold

Let $M$ be a compact connected topological manifold of dimension $n>1$. Suppose the corepresented functor $[M,-]\colon Top_{\ast}\rightarrow Set$ lifts to monoids or equivalently that $M$ is a ...
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Proof of Alexander Duality on Bredon - help with a passage

At page 353 of Bredon's Topology and Geometry, there is stated the Alexander Duality as Corollary $8.7$. I don't understand where does the upper row come from and why is it exact. I thought it was ...
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knot theory: two definitions of equivalence (ambient isotopy and homeomorphism)

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
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Homotopy colimits preserve weak equivalences

It is well known that homotopy colimits of diagrams are constructed so that if one has weak equivalences between all objects of two diagrams (under the same indexing category) the induced map between ...
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what is the classifying space of a monoid

In the paper Homology Fibrations and the "Group-Completion". Theorem. McDuff, D.; Segal, G., 1976, the first line: A topological monoid $M$ has a classifying space $BM$. I do not understand this ...
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Find the fundamental group of $X$

Let $X$ be the unit square with corners identified. I was thinking about its fundamental group. My strategy was to visualize it as e CW complex with a single $0$-cell, four $1$-cells (i.e a wedge of ...
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Formula for Stiefel-Whitney of tensor product

I need help for solving Ex. 7C from 'Characteristic Classes' by Milnor/Stasheff: The exercise asks to find a formula for the (total) Stiefel-Whitney class of $\xi^m\otimes\eta^n$ over a paracompact ...
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$SO(3)$ vs 3-Torus

$SO(3)$ and 3-Torus both can be viewed via rotations for a rigid body. They are not diffeomorphic. $SO(3)$ can be decomposed into three axial rotations. Could I think the reason they are not ...
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Whether or not such a simple CW complex can be made a $C^{\infty}$ manifold?

Problem Let $X$ be the space obtained by attaching two disks to $S^1$, the first disc being attached by the 7 times around,i.e. $z \to z^7$, and the second by the 5 times around. Can $X$ be made ...
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Alexander polynomial of unknot without Fox calculus or infinite cyclic cover

As explained in Lickorish`s book "Introduction to knot theory", one can define the Conway-normalized version of the Alexander polynomial by the determinant of certain sum of Seifert matrix plus ...
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There is no quasiconformal map from punctured ball into ring (on the plane)

The idea is to use l2 cohomology as a quasiregular map invariant. It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form in punctured ...
Can arbitrary cohomology classes $w_1,\dots,w_n$ from $H^{*}(B,\mathbb{Z}_2)$ be Stiefel-Whitney classes of some bundle over the given base $B$ or there are some necessary relation which can be ...