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

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1answer
24 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
0
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0answers
57 views

Prerequisites “Homology Theory of Algebraic Varieties” by Wallace

I bought this book because the title was very interesting, the description as well and the price very cheap. You can read it here in PDF. Unfortunately, I realized after reading the first lines I was ...
2
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0answers
29 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
1
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0answers
53 views

Long exact homology sequence in singular homology

I am trying to understand/develop the proof of the following theorem: Let $R$ be a commutative ring with 1. Suppose $(C_*, c_*), (D_*, d_*), (E_*, e_*)$ are $R$-chain complexes and $i_*: C* ...
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0answers
22 views

Definition of structure group associated with fiber bundle.

I was studying fiber bundle from Spanier book.In that book there is a definition of structure group associated with a fiber bundle.But I am not able to understand the definition properly.Could ...
1
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1answer
56 views

Homology of $S^2/x\sim -x$ for $x$ on the equator

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim -x$ for $x$ in the equator $S^1$. Compute the homology groups $H_i(X)$. I wrote my solution/attempt below and I would like ...
7
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80 views

How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
2
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1answer
25 views

Why is the induced homomorphism an injection?

I am reading Hatcher's Algebraic Topology. One of the propositions says that if a space X retracts to a subspace A, the the homomorphism i# induced by the inclusion i: A --> X is injective. It is ...
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
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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
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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
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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
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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
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1answer
112 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
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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 ...
1
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0answers
32 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
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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
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2answers
71 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
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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
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2answers
71 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
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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
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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
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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
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0answers
13 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 ...
1
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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
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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
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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 ...
1
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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
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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
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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
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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
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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
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
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1answer
42 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
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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
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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
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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
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4answers
179 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
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 $L(p,q)$?
2
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0answers
59 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
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
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0answers
90 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.