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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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Identify all three edges of a single 2-simplex does not produce a Delta-complex structure

Why this does not produce a Delta-complex structure?
Inuyasha
  • 479
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Simplex Triangulation of Cylinder and Mobius Strip

This is an example of triangulation of simplex from the book by M. Nakahara. This example is of an unoriented simplex. It says that $\langle p_0\rangle\cup\langle p_2\rangle$ is not a simplex, why not, they both are points and points are valid…
user3001408
  • 817
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Does isotopy in a covering space imply isotopy in a base space?

If $p:\tilde{X}\rightarrow X$ is a regular covering space of finite degree, why is it not obvious that if two curves $\gamma$ and $\delta$ are isotopic in $\tilde{X}$ their images are isotopic in $X$? By my understanding, this is a nontrivial…
Becca Winarski
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Expressing the sections of the Möbius bundle on $S^1$ as globally defined real valued functions

Consider the real Möbius bundle over $S^1,$ defined as follows. The bundle is trivial over $U_1 = S^1 \setminus \{i\}$ and $U_2 = S^1 \setminus \{-i\}$ and the transition function $T_{12}$ defined on $U_1 \cap U_2$ is given by $T_{12}(z) = 1$ when…
mck
  • 1,276
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finding fundamental group of simplicial complexes

I am currently studying a basic course in algebraic topology.I am finding it difficult to understand the intuition behind the method of finding fundamental group of a simplicial complex using generators and relations. "what is the context of…
abhishek
  • 101
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Fundamental group of $G/H$

Let $G$ be a connected topological group. And $H$ be a discrete subgroup. Theorem : $\pi_1(g/H)=H$ This is the content in the book Algebraic Topology -Greenberg and Harper I want to know the proof. In this book, take an open set $U$ around $1$ such…
HK Lee
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Quotient of a triangle

This is an exercise in Hatcher's Algebraic topology: What familiar space is the quotient $\Delta$-complex of a $2$-simplex $[v_0,v_1,v_2]$ obtained by identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$, preserving the ordering of the edges? The…
Jochen
  • 12,254
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An open set of $ \mathbb{R}^2$ without a point is not simply connected

An open set of $ \mathbb{R}^2$ without a point is not simply connected I need a rigorous proof of this because I only have the intuitive idea that a loop around the point can not be deformed into a point.
davidivadful
  • 462
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How do I prove that $S^1\vee S^2\vee S^3$ and $S^1\times S^2$ are not homotopy equivalent using homology and cohomology ring respectively?

How do I prove that $S^1\vee S^2\vee S^3$ and $S^1\times S^2$ are not homotopic using homology and cohomology ring respectively? They have the same homology groups by Kunneth. There is an exercise in Rotman's algebraic topology to prove that these…
Rubertos
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What is the easiest way to see $\langle \Sigma X, Y \rangle\cong \langle X,\Omega Y\rangle $

Let $X$ and $Y$ be topological spaces. Let $\langle X,Y\rangle$ denote the homotopy classes of maps from $X$ and $Y$. The reduced suspension $\Sigma(-)$ has the adjoint $\Omega(-)$. In other words, we have $$ \langle \Sigma X, Y \rangle\cong…
Tian
  • 51
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If $A$ is a path connected subspace, is $q_*: \pi_1(X) \to \pi_1(X/A)$ a surjection?

Let $X$ be a topological space and $A \subset X$ be a path connected subspace. Let $q: X \to X/A$ be the quotient map. Every example i work out it seems that the induced map $q_*: \pi_1(X) \to \pi_1(X/A)$ is a surjection. Is this always true? Are…
902
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There is no retraction of the solid torus $S^{1} \times D^{2}$ onto the torus $S^{1} \times S^{1}$.

I'm trying to use the fact that I know that there is no retraction of $D^{2}$ onto $S^{1}$ (since there can be no injection $\pi_{1}(S^{1}) \rightarrow \pi_{1}(D^{2})$) to show that if there were a retraction $r:S^{1} \times D^{2} \rightarrow S^{1}…
Tuo
  • 4,556
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Non contractible space with contractible suspension

I know that there exist non-contractible spaces $X \not\simeq \ast$ with contractible suspension $\Sigma X \simeq \ast$. For instance the 2-skeleton of the Poincaré homology 3-sphere is such a space. But is there such a space which is itself a…
toto
  • 51
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Hatcher Problem 1.2.15: Injectivity of an induced map of fundamental groups

I have as an assignment problem that is Hatcher Exercise 1.2.15. The exercise is basically this. Given a space $X,x_0$, we construct a space $L(X)$ built from a one 0-cell, for each loop at $x_0$ in $X$ we have a 1-cell in $L(X)$, and for every map…
user38268
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homotopy groups of a pair and quotient

It is known that if $(X,A)$ is a good pair, for example a $CW$ pair, then $H_k(X,A)\simeq H_k(X/A)$ for every $k$. Is it true for homotopy groups of $CW$ pairs? If not, what is the counter-example?
Morton
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