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Questions tagged [algebraic-topology]

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

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Is it true that $[\overline{g*\alpha}] = [\bar{g}*\bar{\alpha}]$?

$x_0,x_1$ are points in the path-connected space $X$. Is it true that $[\overline{g*\alpha}] = [\bar{g}*\bar{\alpha}]$, where $g\in \pi_1(X,x_0)$ and $\alpha$ is a path from $x_0$ to $x_1$? $g*\alpha$ starts at $x_0$ and ends at $x_1$, so…
user546106
  • 706
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How does induced homomorphism for fundamental groups work in the combinatorial setting

For simplicial complexes, there is a combinatorial approach to defining the fundamental group $\pi_1$, involving maximal trees and generators with relations (see for instance Armstrong's book pg 133-135). This definition does not use the idea of…
yoyostein
  • 19,608
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$\mathbb{R}P^n$ in an attaching space construction of CW complex

Just begin to study the CW complexes and not sure if I understand the way the projective plane is constructed as one. In Topology and Geometry by Glen Bredon it is defined as a quotient of $S^n$, which is constructed via attaching 2 "antipodal"…
apo
  • 35
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How to calculate Bockstein homomorphism?

For a given coefficients sequence, $0 \to Z \to Z \to Z_2 \to 0$, where the map $Z \to Z $ is defined as times 2. My question is how to calculate the Bockstein homomorphism for real projective plane and Klein bottle? Thanks!
Jiabin Du
  • 609
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Flipping the torus coordinates

Here's the path on the torus parametrized by $(t,t/12)$ (i.e. twelve revolutions in one directions for each revolution in the other direction): And here's the same path on a 3D embedding of the torus: Now, if we flip the coordinates (by reflecting…
kjo
  • 14,334
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General Chern number question.

In physics we often define the Chern number as the closed integral over the Berry curvature $$\Omega_{xy}=\frac{\partial A_x}{\partial k_y}-\frac{\partial A_x}{\partial k_x}.$$ With, $$A_i(\mathbf{k})=i\langle…
Jens Roderus
  • 407
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What are two different homotopy classes of loops at one base point?

I am having difficulty in distinguishing between two equivalence classes of the fundamental group at a base pont $x_0$ of a topological space $X$. Given $X$ and an arbitrary point $x_0\in X$ one defines homotopy as equivalence relation on the set of…
user249018
  • 1,480
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Homologies of the pairs are same but they are not homotopy equivalent as pairs.

Consider the pairs $(\mathbb D^{n},S^{n-1})$ and $(\mathbb D^{n},\mathbb D^{n}-\{0\})$ ,clearly their homologies are same in each dimensions but these pairs are not homotopy equivalent. Any homotopy equivalence $f:(X,A)\to(Y,B)$ induce a homotopy…
user51266
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Induced map on homology

If I'm given a map from $\mathbb{C}P^1\times\mathbb{C}P^1$ to $\mathbb{C}P^3$ which sends $([z_{0},z_{1}],[w_{0},w_{1}])$ to $[z_{0}w_{0},z_{0}w_{1},z_{1}w_{0},z_{1}w_{1}]$, how do I compute the induced map on the second homology group $H_{2}$? So…
Ryan Hunter
  • 113
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Degree of maps on the sphere in spherical coordinates

The map $S^2 \to S^2$ is given in spherical coordinates as follows: $$x = \cos(\phi)\cos(\psi),\ y = \cos(\phi)\sin(\psi),\ z = \sin(\phi).$$ $(\phi,\ \psi) \mapsto (n\phi,\ m\psi)$. What is the degree of this mapping? Is it true that this mapping…
Akari Gale
  • 134
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Spectral Sequences and Inverse Limits

I recently learned about spectral sequences (which I am still trying to understand). In class we defined a (co)homologically graded exact couple (of $A_p^n$s and $C_p^n$'s with vertical arrows in the $A$ columns) to be a certain grid of exact…
Vitaly Lorman
  • 1,830
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CW-structure on $S^n$ and orientations

The sphere $S^n$ has a CW-structure consisting of two cells in each dimension, say $S^n=e^0_1\cup e^0_2\cup\cdots \cup e^n_1\cup e^n_2$. But specifying a CW-structure is more subtle than that, am I right? Because for each cell one has to give the…
Marco Flores
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Chain homotopy on linear chains: confusion from Hatcher's book

I am following the book of Hatcher, see page 121-122 for notations. Define $T:LC_n(Y)\rightarrow LC_{n+1}(Y)$ as follows inductively: for $n=-1$ it is zero map. For $n\geq 0$, $$T(\lambda):=b_{\lambda}(\lambda-T\partial\lambda).$$ Question.…
Beginner
  • 10,836
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How to compute relative homology group

How to compute relative homology group $H_n(SO(n),SO(n)-E_n)$ I try to use the long exact sequence as follows $$ ...H_n(SO(n)-E_n)\rightarrow H_n(SO(n))\rightarrow H_n(SO(n),SO(n)-E_n) \rightarrow H_{n-1}(SO(n)-E_n)...$$ I find difficulty both in…
Bruce
  • 181
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Generator of the fundamental group of $ \mathbb RP^{2}$.

Take a closed hemisphere and identify the antipodal points on the equator ,we get $\mathbb RP^{2}$ and inside $ \mathbb RP^{2}$ we have copy of $ \mathbb RP^{1}$.So, what will be the induced map on fundamental group induced by inclusion? I think ,if…
user51266
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