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

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Existence of Shafarevich maps(theorem 3.6) on Kollar 's book

This problems is on Kollar's Shafarevich Maps and Automorphic Forms. The proof of theorems 3.6 ( page 41) need to find a open immersion Let the $D_w$ be closed subset which is not immersion. Why ...
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24 views

Odd maps $S^n\to S^n$ have odd degree

Let $f:S^n\to S^n$ be an odd map, i.e. $f(x)=-f(-x)$. Then $f$ has odd degree. Hint: You may use the fact that there is no odd map $S^n\to S^m$ if $n>m$. I have an idea for how to prove ...
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23 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
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19 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
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1answer
40 views

Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...
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36 views

Fundamental group two torus minus a single point?

So, if I take one torus and take of one single point, what will be its fundamental group? I think that one single point will not change the topology in this sense. Or will? If yes, how?
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23 views

Homotopy Equivalence and Local Coefficient Systems

Suppose I am computing the (co)homology of a nice space M using a local coefficient system G, i.e. $H_{*}(M, G)$. If M is homotopy equivalent to N, then M and N should have isomorphic (co)homology. ...
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1answer
55 views

Question about simply connected spaces.

I am reading Hatcher's Topology and in it, it is noted that a space is simply connected, by definition, if and only if it is path connected and has trivial fundamental group. Can someone provide some ...
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48 views

when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
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1answer
19 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 ...
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51 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 ...
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27 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 ...
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46 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
15 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 ...
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1answer
43 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 ...
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52 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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56 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 ...
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27 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 ...
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1answer
44 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 ...
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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
29 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 ...
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42 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 ...
3
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1answer
82 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
35 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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30 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 ...
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0answers
21 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
49 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
67 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
61 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
63 views

Fundamental Group of a Surface [on hold]

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 ...
3
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0answers
63 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 ...
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1answer
46 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 $|z| = |w| = 1$ ...
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0answers
8 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
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1answer
110 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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41 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 ...
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0answers
30 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 ...
5
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1answer
72 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 ...
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0answers
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 ...
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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
113 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 ...
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0answers
43 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 ...
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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
57 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, ...
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0answers
28 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 ...
4
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1answer
61 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?
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0answers
25 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 ...
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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 ...
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49 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 ...