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

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2
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0answers
63 views

Singular homology: Change of coefficients

Let $f: X \to Y$ be a map of topological spaces which induces isomorphisms $H_*(f;\mathbb{Z})$ on singular homology with $\mathbb{Z}$-coefficients. Show that $f$ induces isomorphisms ...
1
vote
0answers
28 views

A function between covering spaces.

Given $p_1:\bar X_1\rightarrow X$ and $p_2:\bar X_2\rightarrow X$ covering maps. Proof that if exist $f:\bar X_1\rightarrow \bar X_2$ continuos and surjective then $f$ is a covering map. I don't know ...
2
votes
1answer
57 views

How can I get a cohomology of hypersurfaces by using their equation?

While studying about complex projective hypersurfaces, I attempts to find a cohomology of this hypersurface : $$X_n=\{(x_0:x_1:x_2:x_3) \in \mathbb{C}\mathbb{P}^3~|~x_0^n+x_1^n+x_2^n+x_3^n=0\}$$ I ...
1
vote
1answer
29 views

The preimage of a curve in the projective plane by the quotient map.

Let $q:S^n \rightarrow \mathbb{R}P^n$ the quotient map between the $n$-sphere and the $n$-dimensional projective plane. Prove that if $\alpha$ is a curve in the projective plane then $p^{-1}\alpha$ ...
2
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0answers
32 views

Base of homology on a Riemann surface and holomorphic differentials

I have two questions: 1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of ...
1
vote
0answers
43 views

Function from $S^n$ on $S^1$.

Show that for $n\geq 2$ there is not any function $\phi: S^n \rightarrow S^1$ such that $\phi(-x)=-\phi(x) $ I have no idea about how to solve this problem. It is quite similar to Bursuk-Ulam ...
1
vote
1answer
108 views

Some possible mistakes in Bott and Tu

I've been reading Bott and Tu's book "Differential Forms in Algebraic Topology" and it seems that this book has an enormous list of errors. Therefore I would like to confirm if the parts that I will ...
2
votes
1answer
40 views

homomorphism of $H$-spaces between a monoid and loop space of its classifying space

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. The geometric realization ...
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0answers
31 views

spin structures definition

One way to define a spin structure on a $SO(n)$-bundle $p:E\to B$ is to require that there is an element $\sigma\in H^1(E;Z_2)$ such that restricted to each fiber gives a generator of ...
1
vote
0answers
22 views

Computing monodromy eigenvalues of a generic arrangement

First the set-up: Let $f = \Pi_{i=1}^{d} f_{i} \in \mathbb{C}[x,y]$ be a generic, homogoeneous hyperplane arrangement of degree d. Let M be the complement of $f^{-1}(0)$, $F = f^{-1}(0)$, and $p: ...
5
votes
2answers
69 views

Homeomorphisms of the Open Disk

Does there exist a homeomorphism $\phi$ of the open unit disk in the plane such that $\phi$ has no fixed point but there exists $n$ such that the $n$-fold composition $\phi^n$ is the identity? (To ...
4
votes
1answer
73 views

Group action and smooth manifolds

I was wondering if it is for a compact (i.e. Hausdorff) smooth manifold $M$ sufficient to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
3
votes
2answers
70 views

Complete and unabridged proof of the theorem of acyclic models

Can someone indicate me where I can find a complete and unabridged proof of the said theorem? By "complete and unabridged" I mean not writing something like "details are left to the reader as an ...
2
votes
0answers
65 views

Fundamental Group of a Surface [closed]

I came across the term "fundamental group of a surface" while reading a paper, and I'm not sure what it it all about. As well, what is understood by the generators of the fundamental group of a ...
4
votes
0answers
72 views

Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
1
vote
1answer
54 views

Fundamental group of a modified annulus

Let $A\subseteq C$ be the annulus given by $A=\left\{z|1\geq|z|\geq\frac12\right\}$. Define an equivalence relation on $A$ as follows: two different points $z, w$ are equivalent if $|z| = |w| = 1$ ...
1
vote
0answers
12 views

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of L^2?

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of $L^2$? Even elementary clarifications of basic concepts on the filtration side, in ...
5
votes
1answer
116 views

Why is the full subcategory consisting of simply connected spaces not complete?

Let $\mathbf{Top}_*$ be the category of pointed topological spaces and $\mathbf{Top_1}$ the full subcategory of simply connected spaces. $\mathbf{Top}_*$ is complete and cocomplete. I am trying to ...
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0answers
42 views

Inclusion in cone is homotopy equivalence

Suppose $X$ is a topological space and $x \in X$. Let $CX$ be the cone of $X$, i.e. the quotient space $X \times [0,1]/{\sim}$ where $(x,1) \sim (y,1)$ for alle $x,y \in X$. I would like to show that ...
1
vote
0answers
34 views

Is the Poincaré Lemma related to Hatcher's prism operator?

I've been trying for days to understand the statement, content, and proof of the Poincaré Lemma. In hindsight, I think the Poincaré Lemma first appeared (secretly) in my first course in ...
6
votes
1answer
85 views

Proof of Kunneth's formula in Bott & Tu

Let $M, F$ be smooth manifolds and let's assume all (henceforth, de Rham) cohomologies of $F$ are finite-dimensional. Let $\pi : M \times F \rightarrow M$ and $\rho : M \times F \rightarrow F$ be the ...
2
votes
1answer
38 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 ...
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0answers
57 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*} ...
7
votes
1answer
164 views

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

Edit: As there are many comments and an answer already, I have left the original question below. I was unaware that there are different ways one could try to define $\mathsf{hTop}_{\bullet}$, 'the' ...
2
votes
0answers
46 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
votes
0answers
33 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 ...
0
votes
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
votes
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
votes
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 ...
4
votes
1answer
41 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
votes
1answer
44 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
votes
0answers
46 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
vote
0answers
52 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
votes
0answers
26 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 ...
1
vote
4answers
178 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
votes
0answers
54 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
58 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
votes
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
votes
0answers
29 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
votes
0answers
89 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
vote
2answers
61 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
votes
1answer
68 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.
2
votes
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
vote
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
votes
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
vote
1answer
42 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
72 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 ...
0
votes
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
vote
1answer
62 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
vote
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 $X$ ...