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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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Homology of union of two cylinders

Let $X:=\{(x,y,z)\in \mathbb{R}^3: (x^2+y^2-1)(x^2+z^2-9)=0\}$. How do I compute $H_k(X)$? First note that $X$ is homotopic to the following space by deformation retraction. Now, to have a Mayer-Vietoris sequence, chop $X$ into a excisive couple as…
Rubertos
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Chern Classes as invariants

the first Chern class $c_1(-):Vect_\mathbb C ^n(-) \rightarrow H^2(-;\mathbb Z)$ is a complete invariant, i.e. yields an isomorphism of groups when evaluated at $X$ with the homotopy type of a CW complex. I am firstly looking for concrete examples…
TheHumanHighway
  • 261
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Is every subgroup of the fundamental group a fundamental group of a covering space?

Let $X$ be a path-connected space and $H$ a subgroup of $\pi_1(x,X)$ and let $x\in X$ be a point. Is the following true? There exists a covering map $\rho:C\rightarrow X$ and $c\in \rho^{-1}(x)$ such that $\pi_1(C,c)\cong H$. The thing that is…
Asinomás
  • 105,651
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Covering space from $D^{2} \rightarrow S^{1}$?

Is there a covering space $p : D^2 \rightarrow S^1$? I'm not sure how to go about solving this problem. I considered maps such as $z \rightarrow \frac{z+a}{|z+a|}$, but I'm not sure how to show where or not this is satisfies all of the…
Donny Gabo
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Covering map from $S¹$ to $S^1$?

I'm trying to show that the following map is a covering map: $ p_{n}: S^{1} \rightarrow S^{1},\space(\cos(t), \sin(t)) ↦ (\cos(nt), \sin(nt))$ I've taken the approach of dividing $S^{1}$ into four overlapping sections: 1) positive first coordinate,…
bones_mccoy
  • 573
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Is the degree of a mapping on a ball equal to its degree on the ball's boundary?

Each continuous mapping between compact oriented manifolds of the same dimension has a degree, which is an integer number. Let $f$ be a continuous mapping from the $n$-dimensional ball $\mathbb{B}^n$ to itself. Suppose $f$ maps the boundary of…
Erel Segal-Halevi
  • 10,555
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How can we prove that the North hemisphere is homeomorphic to RP²?

How we can prove that the North hemisphere is homeomorphic to the projective plane RP²?
user42912
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Covering space lifting property

Let $p:Y\rightarrow X$ be a covering space and let $x_0 \in X$. Define a map $\pi_1(X,x_0)\to M(p^{−1}(x_0))$ where the last bit is the set of maps from $p^{−1}(x_0)$ to itself. The map sends $[f]$ to the map $\phi_{[f]}(x)=$ end point of the lift…
Bob
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Which groups can be simplicial homology groups of a topological space?

I was wondering about which groups can be the simplicial homology groups of a topological space? For example, is it possible to construct a topological space $X$ such that $H_1(X;\mathbb{Z}) = \mathbb{Z}/3\mathbb{Z}$ and $H_2(X;\mathbb{Z}) =…
mtsecco
  • 113
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Projective space, explicit descriptions of maps.

Consider $\mathbb{P}_\mathbb{R}^2$, i.e. the $\mathbb{R}$-projective plane. I have two questions. My first question is, what is an explicit description of the isomorphism between $H_*^{\text{cellular}}(\mathbb{P}_\mathbb{R}^2)$ and…
user387384
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2 answers

Is there an example of compact nonorientable $n$-manifold s.t. $H^i(M)\cong H_{n-i}(M)$ fails?

Is there an example of compact nonorientable $n$-manifold s.t. $H^i(M)\cong H_{n-i}(M)$ fails ?
6666
  • 3,687
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The property of suspension

THe suspension $\Sigma X$ of a topological sapce $X$ is defined as the qutient space $$ \Sigma X=\dfrac{X\times [0,1]}{\sim}$$ Where $(x,t)\sim(y,s)$ if and only if $s=t=0$ or $s=t=1$ or $(x,t)=(y,s)$. Sow that $\Sigma X$ is simply connected if $X$…
Jack
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Identifying the boundary of a Möbius band to a single point

The Möbius band has the polygonal presentation $$\langle a,b,c \vert abcb\rangle$$ Now I am asked to identify the boundary of the Möbius band to a single point. How do I represent this using polygonal presentations? Is that even possible? I am…
TheGeekGreek
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Compact manifolds having the $\mathbb{R}^2$ as its universal covering

Is there any compact surface having $\mathbb{R}^2$ as its (universal, of course)covering space other than torus and the Klein bottle? While I was thinking of the tiling the plane with a regular polygon, I came to suspect there is only two surfaces…
HyJu
  • 701
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A question about Kronecker pairing

Corollary 2.31. If $R$ is a field, $M$ is a vector space over $R$, and $C_*$ is a chain complex over $R$, then $$H^q\left({C_*;M}\right) \cong \operatorname{Hom}\left({H_q\left({C_*}\right),M}\right)$$ Moreover the Kronecker pairing is…
Danny
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