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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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If the suspension of a topological space X is a n-manifold, why does X have to be a homology sphere?

I am currently working on the following problem : Let $X$ be a topological space such that its suspension $\Sigma X$ is a n-manifold. Show that $H_k(X) \cong H_k(\mathcal{S}^{n-1})$ for all $k$. I already proved that $H_k(X) \cong H_{k+1}(\Sigma…
KnitKnot
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Which perfect groups occur as fundamental groups of finite spaces?

Is there a way of constructing the Hasse diagram of a finite T0-space for a given perfect finite group?
perfin
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Applications for singular homology with coefficients in abelian groups that aren't rings?

Hatcher's Algebraic Topology defines singular and cellular homology taking coefficients in a general abelian group, rather than just $\mathbb{Z}$. However, all of the actual examples seem to use abelian groups that are naturally viewed as rings,…
Justin Toyota
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Euler characteristic of $n$-dimensional disk $D^n$ seen as a CW-complex is -1?

I know that the Euler characteristic of the closed $n$-dimensional disk $D^n$ is $\chi(D^n)=1$ (see e.g. this question). On the other hand, $D^n$ can be seen as a CW-complex consisting of only a single $n$-cell. According to the definition of the…
LarsB
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Does a circle retract to an open arc?

I found the following question on Lima's Fundamental groups and covering spaces: Show that a proper subset of $S^1$ is a retract of $S^1$ if and only if it is a circular arc. I see how a proper closed arc, i.e. the image of a proper closed…
Eric Vaz
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$G$ acts freely on a closed surface $S$ and $S/G$ is orientable. Is $S$ orientable?

Armstrong question 7.8: Let $G$ be a finite group which acts as a group of homeomorphisms of a closed surface $S$ in such a way that the only element with any fixed points is the identity. Show that the orbit space of $S/G$ is a closed…
Christian Chapman
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”figure 8” space embedded in S2

Let M3 be the 3-manifold defined as the quotient space of I × S2 by the identification {0} × {x} s {1} × {Tx}, where T : S2 → S2 is a reflection through a plane in R3. Find π1(M) and π2(M).
midia sharifi
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a closed 1-form which is not exact

On the unit circle S1 in the plane, let θ = arctan(y/x) be the usual polar coordinate. Show that dθ makes sense on S1 and is a closed 1-form which is not exact.
midia sharifi
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Why is the induced orientation of a simplex well-defined?

In my algebraic topology class I learned that let $A = (v_0,\cdots, v_k)$ be an oriented simplex and $B$ the face of $A$ by removing the vertex $v_i$, then the orientation for $A$ induces an orientation for $B$, namely $$ (-1)^i (v_0,\cdots,…
Coco
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The definition of a contractible space.

I wonder if my approach is completely wrong. If so, may I request for some hints for heading to the right direction? Thank you! Show that a retract of a contractible space is contractible. The previous discussion Proof that retract of…
1LiterTears
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Decomposition of a separating curve

Consider a genus 2 surface. Its fundamental group can be expressed by $$ \pi_1(S_2)=\langle\alpha_1,\beta_1,\alpha_2,\beta_2\:||[\alpha_1,\beta_1][\alpha_2,\beta_2] \rangle. $$ I would like to know how to express the curve $\gamma$ as product of…
Erick Gordillo Herrerías
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Restriction of sections a weak homotopy equivalence?

Suppose we are given a fiber bundle $p:E \to B$ and a point $x \in B$. Denote by $p \big|_{p^{-1}(x)}:p^{-1}(x) \to B$ the restricted fiber bundle and by $\Gamma^0(p)$ (resp. $\Gamma^0(p \big|_{p^{-1}(x)})$) the space of continuous sections with the…
Martin Worsek
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A question about the proof that a homotopy equivalence induces an isomorphism

If $f:X \rightarrow Y$ is a homotopy equivalence with homotopy inverse $g$ and I want to prove that $f_*$ is an isomorphism then Hatcher (on page 37) uses the following: $$ \pi_1 (X, x_0) \xrightarrow{f_*} \pi_1 (Y, f(x_0)) \xrightarrow{g_*} \pi_1…
Rudy the Reindeer
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Find the Fundamental group of this space

Let X be the space obtained from $S^2$ by identifying (x; y; 0) with (-x; -y; 0), for all (x; y; 0)$\in$ S2. Compute $\pi_(X).$. I know to choose the open sets so that they each deformation retract to th real projective plane and the intersection…
Eddie
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How can one prove that circle isn't a retract of a ball without using Homotopy or Homology theory?

Are there some "elementary" ways to prove that circle can't be expressed as a two dimensioanl disks retract?
ალექსანდრე ჭაღალიძე
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