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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.

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21356 questions
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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 showing the Cayley complex for…
PeterM
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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. The quotient space $X$ will look like this, isn't…
Xena
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Reverse use of Seifert-van Kampen Theorem?

I am trying to use S-vK Theorem in reverse; what I know are as follows: $U$ and $V$ satisfy the requirements (open, path-connected), $U\cup V = X$, $U \cap V = N$ $\pi_1(N) = \langle c,d| cd=dc\rangle$ $\pi_1(U) = ??$ $\pi_1(V) = \langle…
wilsonw
  • 1,016
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Creating connective spectra from infinite loop spaces

I have a quick question which I think should go like this, but I am not really sure and that is why I would like someone more knowledgeable than me to weigh in and say if I am correct. Let us say that $X$ is an infinite loop space - it is well-known…
Tedar
  • 529
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Injectivity between non-trivial knot on torus and $S^1$ on torus.

$X = S^1 \times D^2$ and $A$ the circle shown in the figure, Show that there are no retractions $r \colon X \to A$. Assume for contradiction that there is a retraction $r \colon X \to A$, then that means $$i_*\colon \pi_1(A, x_0) \to \pi_1 (X,…
1LiterTears
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Why is $D^{n+1}/S^{n} = S^{n+1}$ true?

I went to my first lecture in Algebraic Topology and managed to get really confused. It seems like they assumed that the following statement was "obvious": $D^{n+1}/S^{n} = S^{n+1}$ Where $D^{n}$ is the unit disk/ball in $\mathbb{R}^{n}$ and $S^{n}$…
user93024
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Doubt from Allen Hatcher's AT about CW Complex

I am trying to learn Cellular Homology from Allen Hatcher's AT book, but stuck in first Lemma (2.34) itself. While introducing Cellular Homology Hatcher in his AT, Section 2.2 Lemma 2.34 says $X^n/X^{n-1}$ is a wedge sum of $n$ - spheres, one for…
Ram
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Streamlining of Hatcher's Algebraic Topology Proposition 1.40 sufficient conditions

In Hatcher's Algebraic Topology Proposition 1.40, it is said that, for a covering action of a group $G$ on a space $Y$, (b) $G$ is the group of deck transformations of the covering map $Y\to Y/G$ if $Y$ is path-connected. It seems to me that $Y$…
brunoh
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Example of an oriented manifold with cohomology not isomorphic to a homogeneus space

The question as in the title: Is there a simple example of a compact orientable smooth finite-dimensional manifold whose singular cohomology groups with integer coefficients are not isomorphic to those of some smooth homogeneous space? My heart…
BebopButUnsteady
  • 505
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How to give the coproduct of differential graded algebras explicitly?

Let $X$ and $Y$ be based spaces such that their respective loop spaces $\Omega X$ and $\Omega Y$ are connected. In the first paragraph of this article by Dula and Katz, it is given that $H_*(\Omega(X\vee Y))=H_*(\Omega X) \coprod H_*(\Omega Y)$…
user17982
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What is the range of the fundamental group?

Possible Duplicate: is the group of rational numbers the fundamental group of some space? Given any group G,can we find a complex X whose fundamental group is G.If not,which kind of group satisfies this condition?
Strongart
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In Hatcher's proof that two homotopic maps induce the same homomorphism in homology what do the prism operators represent?

On page 112 in the proof of Theroem 2.10 Hatcher defines the prism operators given by: $P(\sigma)=\sum_i(-1)^iF\circ(\sigma\times Id_I)\mid[v_0,\cdots,v_i,w_i,\cdots,w_n]$ $F\circ(\sigma\times Id_I):\Delta^n\times I\rightarrow X \times I \rightarrow…
Dan Disney
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showing a map is not null-homotopic

How do I show that two maps are not homotopic. More precisely, how do I show that a map is not null-homotopic. 1-If $f,g:S^n \to S^n$ then its degree should be same. if their degree isn't same then they aren't homotopy equivalent. 2-Since homotopic…
permutation_matrix
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Why does this particular Serre spectral sequence collapse at the $E_{k+2}$-page?

The problem: Let $X$ be a product of equidimensional spheres of arbitrary dimension, say $k$, and $G$ a finite group acting freely on $X$. Assume that the induced $G$-action on the $Z_2$-cohomology ring of $X$ is trivial. We then have a Serre…
swan
  • 163
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How does a group action on a space induce action on cohomology ring of the space?

This is mostly a question to make sure I have some signs correct. Let a group $G$ act on a space $X$. As notation, for $g \in G$ let $\phi_g : X \rightarrow X$ be the map induced by the action of $g$. It is then clear that we get an action on the…
Mike
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