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

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1answer
85 views

Different versions of Hatcher

I suddenly found out that my Hatcher from amazon is very different from the version on his website. Should I assume his website is up to date, and hence my copy is an old version?
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2answers
85 views

what is the meaning of a symbol $\pi_1(X,x_0)=0$

what is the meaning of a symbol $\pi_1(X,x_0)=0$ when $X$ is a contractible space to $x_0$. Actually I know that $\pi_1(X,x_0)$ is the fundamental group of $X$ based at $x_0$.But I could not ...
1
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2answers
69 views

Some questions on the definition of $n$-simplex.

Hatcher P102 last paragraph: $n$-simplex is the smallest convex set in $\mathbb{R}^m$ containing $n+1$ points $v_0, \dots, v_n$ that do not lie in a hyperplane of dimension less than $n$. I ...
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1answer
41 views

$a - b, b-c, c-d$ form a basis for this kernel.

Hatcher P99 last paragraph: Define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of the edge minus the vertex at the tail. Thus ...
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13answers
2k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
3
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0answers
58 views

Serre spectral sequence and locally constant coefficients

I have a brief question - In the Serre Spectral sequence for a fibration $$F \rightarrow E \rightarrow B$$ one can require, to avoid using local system of coefficients, that the action $\pi_1(B)$ on ...
7
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3answers
548 views

Mapping homotopic to the identity map has a fixed point

Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is a fixed point $\phi(p)=p$.
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1answer
35 views

Topological interpretation of a zero map.

I have a pair of complexes of modules $A_{\bullet}$ and $B_{\bullet}$, and I want to create a new complex pretty trivially by shifting one complex by 1, and then taking the direct sum of the ...
1
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1answer
67 views

Modification of Excision Theorem

In the Excision Theorem, there is a condition that the closure of U is contained in the interior of A. Now I wonder if the Excision Theorem is still true when this condition is replaced by the ...
1
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1answer
11 views

Sending each basis element $a,b,c,d$ to $y-x$.

Hatcher P99 last paragraph: Define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of the edge minus the vertex at the tail. So I ...
3
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1answer
572 views

Fundamental Polygon of Real Projective Plane

Wikipedia gives the following fundamental polygon for the real projective plane $\mathbb{R}\mathrm{P}^2$ The problem here is that the corners aren't identified to a single point (like in the ...
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2answers
120 views

to understand a theorem for fundamental group

I faced a problem to understand the proof of the following theorem from the book "algebraic topology by satya deo". If $F\colon X\to Y$ be a homotopy between two maps $ f,g\colon X\to Y $. Let ...
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5answers
185 views

Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
4
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1answer
710 views

How to show the standard $n$-simplex is homeomorphic to the $n$-ball

I am trying to show the standard $n$-simplex is homeomorphic to the $n$-ball. Here, the standard $n$-simplex is given by $$\Delta^n=\left\{(x_0,x_1,\cdots,x_n)\in\mathbb{R}^{n+1}:\sum ...
3
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2answers
149 views

Help with constructing homeomorphism for this identification

Consider the following triangles: I have shown that $T$ is homeomorphic to as disc. Here is the proof: First note that one can prove the following theorem: The mapping $f^\ast : X/\sim_f \to Y$ ...
1
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1answer
252 views

Retraction of surface of genus $g$

This is an exercise in 53 page of Hatcher's book : Consider surface of $M_g$ of genus $g$ If $g=h+k$ then $$M_g = M_h'\cup_{S^1} M_k'$$ where $M_h' = M_h - D^2$ Question 1 : Then show that $M_h'$ ...
6
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2answers
412 views

Cellular Boundary Formula

In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta} $ where $d_{\alpha\beta}$ is ...
3
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0answers
373 views

Galois correspondence of covering spaces of spaces not necessarily semilocally simply-connected

I've been trying to solve the following exercise (1.3.24) from Hatcher's Algebraic Topology: Given a covering space action of a group $G$ on a path-connected, locally path-connected space $X$, ...
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0answers
50 views

Why Is the Induced Map Not Zero?

I am reading "Modern Classical Homotopy Theory" by Strom and have come across the following. We are given a fibration $F\rightarrow E\rightarrow B$. One then has two pushout squares: ...
2
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1answer
329 views

The Stiefel-Whitney classes of Cartesian product

I am reading the book of characteristic classes by Milnor-Stasheff, and I have a problem with the exercise 4-A: Show that the Stiefel-Whitney classes of a Cartesian product are given by ...
4
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1answer
57 views

The Hopf invarient with coefficients other than Z.

So generally one defines the Hopf invariant of a map $f: S^{2n-1} \to S^n$ as the coefficient $H(f)$ in $\alpha^2 = H(f) \beta$ where $\langle \alpha \rangle = H^n(C_f)$ and $\langle \beta \rangle = ...
1
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1answer
284 views

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ...
1
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1answer
83 views

Question on functors

please i need help,how to prove that "the functor (covariant) "fundamental group", of the category of pointed topological spaces in the category of groups" is really a functor What i must do to ...
2
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1answer
41 views

Is this also a homotopy

If $F$ is a path at a point $x$ then the following defines a homotopy from the path $FF^{-1}$ to the constant path $e$: $$ \begin{array}{cc} H(t,s) = F(2t) & s \ge 2t \\ H(t,s) = F(s) & s \le ...
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2answers
104 views

Map induced by $O(n)\hookrightarrow U(n)$ on homotopy groups

There is an inclusion $O(n)\hookrightarrow U(n)$ which views an $n\times n$ orthogonal matrix as a unitary matrix. It is also a theorem, sometimes called Bott periodicity, that we have the following ...
2
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0answers
180 views

Homology groups (?) of some quotient of $\Bbb{R}P^n$

Here is a question from Hatcher (2.2.19): I assume that those $H_i$'s are homology groups, Hatcher denotes both the chain complexes and the homology groups by $H$ (I will denote the chain complexes ...
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1answer
107 views

Simply connected subset of $\mathbb R^3$

Let $C$ be the closed unit cube in $\mathbb R^3$, and let $A$ be one face of the cube $C$ (say the face above and parallel to $xy$-plane). Let $U\subset\mathbb R^2$ be open and path-connected such ...
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0answers
40 views

Finding a homotopy map

Let $K=\mathbb R^2\times (-\infty,0)\subset \mathbb R^3$, and let $Q$ be an open connected subset of $\mathbb R^2$. Is the fundamental group $\pi_1(Q\times [0,1)\cup K)$ trivial? And is it possible ...
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0answers
107 views

Question About Transgression

I have been working on this question here. Here is the setup: First, all cohomology groups are assume to be with $\mathbb{Q}$ coefficients. We assume that $H^*(K(\mathbb{Q},n))=\mathbb{Q}[x]$, with ...
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1answer
108 views

More interesting examples of spaces that are retractions?

I learned about retraction: A continuous map $r: X \to A$ where $A$ is a subspace of $X$ is called a retraction if $r|_A = id_A$. I made some examples. For example: If $D$ is the closed unit disk and ...
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2answers
760 views

Function doesn't have a lift in a space related to Topologist's sine curve

I'm trying to solve exercise 1.3.7 in Hatcher's Algebraic Topology: Let $Y$ be the quasi-circle that is the union of a portion of the graph $y = \sin(1/x)$, the line segment $[-1,1]$ in the ...
2
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0answers
408 views

Homology of nonorientable surfaces

Let $N_g$ be a closed nonorientable surface of genus $g$. I will try to compute the homology groups and I want you to help me with certain steps and correct my mistakes - I will use this as an ...
2
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0answers
48 views

Rational Elienberg-Maclane Spaces

Is it true that $$ H^k(K(\mathbb{Z},n);\mathbb{Q})\cong H^k(K(\mathbb{Q},n);\mathbb{Q}) $$ for all $k$?
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0answers
63 views

Seemingly serious problem with Deformation Retraction

So a friend and I are arguing over Deformation retraction. Any help to settle this would be nice. Consider a T shaped subspace of $\mathbb{R}^2$. Let $A$ be the vertical segment and let $B$ be the ...
2
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1answer
67 views

Collapse of a subspace - Cofibration

Let $i:A \rightarrow X$ be a (closed ) cofibration (i.e a cofibration in the Strøm Model structure). For a subspace $B \subset A \subset X$, when is it true that $A/B \rightarrow X / B$ is a ...
11
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1answer
140 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
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2answers
518 views

The degree of antipodal map, composition of reflections?

Here is a bit from Hatcher's book: I don't understand part (f); why is the antipodal map the composition of $n+1$ reflections? Even if I accept that, I still don't know why does it have degree ...
2
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2answers
324 views

When to attach 2-cells in Cayley complexes?

In Hatcher's Algebraic Topology section 1.3, Cayley complexes are explained. The book states that we get a Cayley complex out of a Cayley graph by attaching a 2-cell to each loop. There is an example ...
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2answers
44 views

A nonseparation theorem for arcs on $S^{2}$

Let $A$ be a simple unclosed curve in $S^{2}$. Is there a simple way to prove that $S^{2}\setminus A$ is path-connected using homology? By simple unclosed curve I mean that $A:I\rightarrow S^{2}$ is ...
3
votes
1answer
281 views

Cohomology groups of a homotopy fiber

I am reading the following: http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf and on page 316 there is a thing that gets me confused: Consider the following situation: Assume that we know that ...
3
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1answer
211 views

Formula for Euler characteristic for quotient space of a CW complex

I know that there is formula for Euler characteristic: $$\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$$ Is there any formula that links between (for CW complex) some complex, subcomplex and quotient ...
2
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1answer
132 views

Rick Miranda exercise complete intersection curve. Prove it and find genus.

The book by Rick Miranda asks to prove that the curve in $\mathbb{P}^3$ defined by the two equations $x_0x_3=2x_1x_2$ and $x_0^2+x_1^2+x_2^2+x_3^2=0$ is a smooth complete intersection curve. Also asks ...
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0answers
49 views

Build a moduli space using homotopy theory.

I'd like to build ''with hands'' the moduli space of stable principal $U(1)$-bundles over a Riemann surface $M$ of genus $2$. I have the follow idea: We know that the isomorphirsm classes of principal ...
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1answer
71 views

Topology question??

Suppose we have a surface $S$. Also, suppose we remove $2$ discs from the surface $S$ and we glue the boundary circles of these two discs together. Is the result a surface?? My believe is that it is ...
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1answer
35 views

Homeomorphism preserving partitions

Let $X$ and $Y$ are homeomorphic topological spaces. Consider a equivalent relation $R_X$ and $R_Y$ that partition $X$ into $X_1,\ldots,X_n$ of and $Y$ into $Y_1,\ldots,Y_n$ respectively. $X_i$ is ...
4
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2answers
480 views

HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
1
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1answer
57 views

Why is this map continuous? (Cofibrations)

Assume that we have a map $i : A \rightarrow X$ which is a (closed) cofibration and a homotopy equivalence. Then, $A$ is a strong deformation retract of $X$ and there is a function $u: X \to I$ such ...
3
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1answer
403 views

Homology of some quotient of $S^2$

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim-x$ for $x$ in the equator $S^1$. I want to compute the homology groups $H_n(X)$. I've seen this but didn't really understand. ...
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1answer
103 views

Help with Cohomologies and Homologies

My algebraic skills are very weak, so in answering please assume I know close to nothing about algebra, geometry, forms, or the like. I am trying to compute homologies and cohomologies. For ...
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1answer
761 views

Find the fundamental group of torus with two points removed [duplicate]

I'm trying to find a fundamental group of Torus \ {two points}. Any help would be really appreciated.