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

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31 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
1
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0answers
54 views

the geometric realization of a simplicial set is contractible

Let $M$ be a monoid up to homotopy. The simplicial set $WM$ is defined by setting $$ WM_n=M^{n+1}=\{(g_0, g_1,\cdots,g_n)\mid g_i\in M\} $$ with faces and degeneracies given by \begin{eqnarray*} ...
6
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1answer
122 views

Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$

Throughout, $(X, x_0)$ and $(Y, y_0)$ will be connected pointed topological spaces. If $f : (X, x_0) \to (Y, y_0)$ is a continuous map and $f_* : \pi_n(X, x_0) \to \pi_n(Y, y_0)$ is an isomorphism ...
2
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0answers
44 views

Why are chains topologically analoguous to distributions?

This question is related to my other question here but is different enough that I thought I might ask separately. At the nLab page on rational homotopy theory it is stated that chains are ...
2
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0answers
30 views

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
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2answers
58 views

Show that $X$ is homeomorphic to exactly one of the spaces in the following list: $S^2, P^2, K, T_n, T_n\#P^2,T_n\#K, n > 0$

Where X is a space obtained by pasting the edges of a polygonal region together in pairs. Alternatively: Show that X is homeomorphic to exactly one of the spaces in the following list: $S^2,T_n, ...
5
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0answers
30 views

Topological invariants that detects change of orientation for an odd dimensional manifold.

Assume all manifolds involved to be closed and orientable. During my studies I learnt about the signature of a $4k$-manifold, and it turned out that (when it is non zero) it can be use to make a ...
5
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1answer
71 views

Is $f$ necessarily a homotopy equivalence? [closed]

Let $X = \{(p, q): p \neq -q \} \subset S^n \times S^n$. Define a map $f: S^n \to X$ by $f(p) = (p, p)$. My question is as follows: is $f$ necessarily a homotopy equivalence? EDIT: Here is my ...
3
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0answers
26 views

Integral of generator $H^{2k}(\mathbb CP^k)$ over $\mathbb CP^k$ is 1

I'm trying to give a proof of the Hirzebruch signature theorem from a differentiable viewpoint (i.e. using de Rham cohomology). The proof works, except for one small step. We know that ...
0
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1answer
42 views

Calculate the first homology group of $P^2\#T$, that is $H_1(P^2\#T)$

I've already found that the fundamental group of the connected sum $P^2\#T$, by the labelling scheme $aabcb^{-1}c^{-1}$, to be $F_3/<aabcb^{-1}c^{-1}>$. How would I find the first homology ...
3
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0answers
44 views

Relations between the homotopy class and the orientation of the connected sum of two manifolds

I want to find some general arguments about why it can happen that $$ M \sharp \overline{M} \not\simeq M \sharp M$$ for $M$ a compact orientable odd dimensional manifold, which is chiral, i.e. ($M ...
1
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0answers
51 views

Relationship between relative homology and reduced homology.

Prove for any homology theory ($H_*, \partial_*$) with values in R-mod that satisfies the dimension axiom there is an isomorphism $H_n(X,A)$ $\cong$ $\tilde H_n(X/A)$ where $A$ $\subset$ $X$ is a ...
2
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0answers
25 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
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4answers
169 views

Applications of algebraic topology?

Terribly sorry if this has been asked, but I'm not about to search 382 pages of technical questions in the field. I am trying to develop a very basic understanding of what algebraic topology is ...
4
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0answers
52 views

3-manifolds with certain fundamental group

Suppose $M$ is a 3-manifold with $\pi_1(M)=\mathbb{Z}/p\mathbb{Z}$. Is it true that $M$ is diffeomorphic to a lens space $L(p,q)$?
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0answers
54 views

prove/disprove $\Delta$ is strongly connected.

Let $\Delta$ be a simplicial complex and $F_1,...,F_n$ be the facets of $\Delta$. Let $\Delta_1$ be another simplicial complex and $F_1,...,F_{n-1}$ be the facets of $\Delta_1$. Assume $\Delta$ and ...
2
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0answers
22 views

pontrjagin ring of the homology of iterated loop suspension

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243) I totally do not understand the proofs of these two theorems from page 228 to page 243 ...
3
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0answers
27 views

Transfer homomorphism in transformation groups

I am aware of the existence of a transfer homomorphism in the setting of so called "regular $G$-complexes", as described e.g. in Bredon's Introduction to Compact Transformation Groups. But suppose ...
4
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0answers
87 views

Mapping $f: (S^n, pt) \to (S^n, pt)$ induces multiplication by $deg(f)$ on $H_*(S^n \wedge X, pt)$.

I'm studying for my final exam in Algebraic Topology and got stuck with somewhat straightforward looking preparation problem. Let $\tilde H$ be any reduced homology theory and $f: (S^n, pt) \to ...
1
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2answers
58 views

Polygonal presentations: why no two-letter words?

In Lee's book Introduction to Topological Manifolds, he discusses polygonal presentations of surfaces. He does so by means of words $W_1, \dotsc, W_n$ such that each letter that appears must appear ...
3
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1answer
61 views

Fundamental groupoid of a contractible space

I read that the fundamental groupoid of a contractible space is indiscrete. How can one show this? I found this as an exercise here.
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0answers
54 views

Show $L_{3,1}\sharp L_{3,1}$ and $L_{3,1}\sharp \overline{L_{3,1}}$ are not homotopy equivalent [on hold]

Let $L_{p,q}$ with $(p,q)=1$ the usual Lens space, I must show that $L_{3,1}\sharp L_{3,1}$ and $L_{3,1}\sharp \overline{L_{3,1}}$ are not homotopy equivalent using homology/cohomology tools. Here, ...
2
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0answers
14 views

Elementary Cube vs. Elementary Chain

I am reading Computational Homology by Kaczynski, Mischaikow, and Mrozek. On page 47, for every elementary cube, $Q \in \mathcal{K}_k^d$ they associate an object $\widehat{Q}$ that they call an ...
1
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1answer
38 views

The fundamental group of a plane without a finite number of points.

How can I calculate de fundamenta group of $\mathbb{R}^2$ without a finite number of points? I know that the answer should be $S^1\vee S^1 \vee \cdots \vee S^1 $ where the product repeats as many ...
2
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0answers
41 views

Deck transformation

I read that a deck transformation is uniquely defined by the value of one point. Unfortunately, I don't understand where this comes from. I mean, all we know is that there is one point in the fibre ...
1
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1answer
36 views

What may be the number of connected components of this randomly generated topological space?

I hope there exists some answer to this but I am not very good in probability so i do not know how to tackle this problem. It's contemporary-art related so it might not have a good answer. ...
4
votes
1answer
68 views

Fundamental polygon square $abab$

What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get ...
1
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0answers
16 views

First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?

In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...
0
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0answers
26 views

Winding number of $S^1$ vector fields with $|u| > |v|$

Let $u$ and $v$ are nonvanishing vector fields on $\mathbb{S}^1$ and $|u(z)| > |v(z)|$ at every point of $\mathbb{S}^1$. Prove that $deg(u) = deg(u + v)$. My idea is to take a homotopy $h_t(z) = ...
1
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1answer
61 views

Properties of Pushout

suppose we have a pushout square in $\mathrm{Top}$: \begin{align*} \require{AMScd} \begin{CD} X_0 @>{\mu_1}>> X_1\\ @V{\mu_2}VV @VV{\alpha_1}V \\ X_2 @>>{\alpha_2}> X ...
1
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1answer
48 views

Homeomorphy of a surface

I am studying graphs on surfaces (i.e. maps). Their definition is below: We call map a representation $(X,\mathcal{D})$ of a finite connected graph $\Gamma=(V,E)$ in the topological surface $X$ ...
4
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1answer
43 views

Does there exist a double cover with trivial deck transformation group?

Sorry for the naive question. The following statement at the beginning of Bredon, chapter 4, §20, got me confused: Let $\pi:X \to Y$ be a two-sheeted covering map. Let $g:X \to X$ be the unique ...
3
votes
1answer
111 views

Can Somebody Please Outline a Reading Course For Me in Algebraic Topology

I want to start self studying algebraic topology and I am looking for guidance regarding the same. In the past I have made the mistake of trying to learn a mathematical subject by reading fat books ...
0
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2answers
66 views

Homotopy/fundamental group question: Why group axioms fail when defined on paths?

Neither Munkres nor Lee in their textbooks explicitly show why (fundamental) group properties like associativity fail when defined at the level of paths but work fine for homotopy classes of paths. ...
2
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0answers
36 views

Calculating the fundamental group of the Klein Bottle using the Seifert-Van Kampen theorem

I want to calculate by two different way the fundamental group of the Klein Bottle. First one: I want to use that the Klein Bottle is can be decomposed in two Mobiüs Band as the following picture ...
2
votes
1answer
46 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
1
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1answer
27 views

What is a linear embedding from a simplex $\Delta^n \to \mathbb{R}^n$?

As stated in the title, reading Milnor-Stasheff Characteristic classes, I encountered at page 95 the following sentence: let $\Delta^n$ be an $n$-simplex, linearly embedded in the $n$-dim vector ...
5
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0answers
69 views

Twisted nail puzzle framed in terms of algebraic topology?

See here for a description of the puzzle: Twisted Nail Puzzle. My question is, can someone provide a description of the puzzle and its solution in context of the language of algebraic topology?
2
votes
1answer
39 views

question about “Homology fibrations and the group completion theorem”

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18: we have a fibre bundle $M_\infty\to (M_\infty)_M\to BM$ with $(M_\infty)_M$ constractible. In ...
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0answers
27 views

Compute the singular homology group of a “rational optical grating”

Let $X$ be the subspace of the square $I \times I$ consisting of the four boundary edges plus all points in the interior whose first coordinate is rational. Calculate the singular homology ...
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0answers
28 views

Lifting of certain $S^1$ valued maps

It's well-known that every path $s(t): I \to S^1$ has a lifting, i.e. mapping $\widetilde{s(t)}: I \to \mathbb{R}$, so that $e^{i\widetilde{s(t)}} = s(t)$. The main idea of constructing ...
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0answers
51 views

Calculate the fundamental group of $S^1/\mathbb Z_n$

Calculate the fundamental group of $S^1/\mathbb Z_n$ ,where $\mathbb Z_n$ acts naturally on $S^1$ by rotations of $2\pi /n$ The origin of this problem is the following unclear solution of another ...
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0answers
53 views

homology of mapping telescope of a monoid

Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ ...
2
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1answer
85 views

A map $h:S^1\to X$ Induces a Trivial Homomorphism of Fundamental Groups Iff it is Nullhomotopic.

I recently started reading Algebraic Topology from Part II of Munkres' book Topology(Second Edition). A part of Lemma 55.3 in the book proves the following: Let $h:S^1\to X$ be a continuous ...
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2answers
73 views

Lists of the first fundamental group of spaces. [closed]

Here are some list to start with $$\begin{array}{c|c|c|} \hline Space(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& ...
1
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1answer
72 views

The winding number (topological definition) is well-defined

can someone please tell me if my proof of the lemma below is legit. At first I like to give a definition so you know what the lemma is about. Definition: Let $g:[0,1] \to S^{1}$ be a closed path in ...
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0answers
53 views

Computing the cohomology of the pair $(S^n\times S^n,D)$

Let $D=(x,x)\subset S^n\times S^n$ be the diagonal, and assume $n$ is even. I need to prove that the following sequence (taken from l.e.s of the pair) is exact $$0 \rightarrow H^n(S^n\times ...
1
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0answers
38 views

configuration space model for classifying space of monoid

Let $M$ be a monoid and $BM$ be its classifying space. There is a model for $BM$ based on labelled configuration spaces of the line $[0,1]$. Points of the configurations are labelled by elements of ...
2
votes
2answers
74 views

Relationship between homology of suspension of $X$ and $X$

The exercise is the following: Show that, for any homology theory (satisfying the usual axioms), there is a natural isomorphism $ \tilde{H_i}(X) \rightarrow \tilde{H}_{i+1}(\Sigma X)$. Well, I ...
0
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1answer
45 views

Representations of Fundamental Group and Monodromy

I have two representations of the fundamental group and I am under the impression they are the same. Any help in seeing this would be great. Preliminaries: Let $\phi: E \to M$ be a n-fold covering ...