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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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Proving that $X = A \cup B$ is a disconnection then $H_n(X) \cong H_n(A) \oplus H_n(B)$ using excision

If $X = A \cup B$ is a disconnection then $H_n(X) \cong H_n(A) \oplus H_n(B)$ If I use the Mayer-Vietoris sequence I can prove this easily, however the book I'm reading hinted at using excision to prove this. So I tried to prove this using…
Perturbative
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Computing the homology of a subspace of $R^3$

Let $P_1 ,...,P_5$ be mutually distinct planes in $\Bbb{R}^3$ such that: the intersection of any distinct 2 is a line. the intersection of any distinct 3 is a point. the intersection of any distinct 4 is an empty set. Let $X=\bigcup_{i=1} ^5 P_i$…
user61974
  • 43
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1 answer

Inverse limits of relative homology in Euclidean space

Let $C$ be a compact set in Euclidean space and $i$ an integer. Is the inverse limit $$\underset{C\subset U}{\lim_\leftarrow}H_i(U,C)$$ over all open sets $U$ containing $C$ of relative homology groups always zero?
Alexander
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Approximating triangles by squares

I have a simplicial complex $X$ embedded in $\mathbb{R}^n$ s.t. $|X|$ is an $n$-manifold. I would like to algorithmically construct a finite set $\Omega$ of $n$-cubes ("cube"=product of intervals) s.t. $\cup\Omega$ deformation retracts to $|X|$. Any…
Peter Franek
  • 11,522
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A question about the $2\times2$ matrices action on 2-torus.

Given a $2\times2$ matrix $A$ with entries in $\mathbb Z$, it acts on \begin{equation} \mathbb T^2=\mathbb R^2/\mathbb Z^2 \end{equation} and then leads to a map \begin{equation} A_*\colon H_2(\mathbb T^2)\to H_2(\mathbb T^2) \end{equation} I want…
Display Name
  • 1,373
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3 answers

Fundamental group of Klein Bottle generated by two elements

The problem asks me to show that the fundamental group of the Klein bottle is generated by "latitudinal" loops $a$ and "longitudinal" loops $b$ where $a$ and $b$ obey the relation $aba^{-1} = b^{-1}$. The problem is that I don't understand what I'm…
user61642
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Example for not deformation retraction but retraction

My professor introduce following example to explain the difference of retraction and deformation retraction. Consider $X=B^2$. Let $A=${lower part of $B^2$} I can find $A$ is retract of $X$, by giving projection map on upper part and identity map…
fivestar
  • 919
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How to prove CP^1 homeomorphic two-dimensional sphere

I'm a complete newbie to this section: book are quite complicated in terms. Also, there is a problem with formal building homeomorphism. So, I need to prove that $CP^1 \simeq S^2 $. How to do it formally? By the way, as I understand, that $RP^1…
DumbSimon
  • 43
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2 answers

How to find that two circles form a knot?

Suppose $f$ and $g$ are two embeddings of $S^1$ in $R^3$ or in $S^3$. How do I show whether they form a link or not?
user8774
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0 answers

De Rham cohomology

I have some question on De Rham cohomology: the first one is general. If we calculate De Rham cohomology of a manifold with Mayer-Vietoris sequences we discover that the cohomology is the difect of exactness of the sequence. So can I look at De Rham…
ArthurStuart
  • 4,932
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1 answer

Does there exist a compact manifold with boundary $M$ that is contractible but does not (strong) deformation retract onto a point?

In general, there exists strange topological spaces that are contractible but do not deformation retract onto a single point. I conjecture that if you have a nice enough space like a compact manifold with boundary that the notion of contractible and…
Tuo
  • 4,556
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3 answers

Fundamental Group of complement of Integers

Let $X=\mathbb{R}\times (-1,1)-\mathbb{Z}\times \{0\}$. I want to compute the fundamental group of that space. I would be tempted to do some division like the following to apply Van Kampen, but to do it all the sets must intersect in the basepoint…
Nell
  • 888
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1 answer

Geometrical interpretation of regular covers

What is the geometrical picture of a regular cover of a topological space, $X$? A regular cover of $X$ being a covering space $(Z,p)$ of $X$ such that, the projection of the fundamental group of $Z$ under the obvious homomorphism is normal in the…
Temari
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Does the Identity Map on $S^2$ Extend?

We form $X$ by attaching a $3$-cell to $S^2$ by a degree $2$ map, and form $Y$ be attaching a $3$-cell to $S^2$ by a degree $4$ map. I am trying to find out whether the identity $S^2 \to S^2$ extends to a map $Y \to X$. I have already showed that it…
J126
  • 17,451
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Torsion in a manifold embedded in the 3 sphere

This is an old qual question. Let $K$ be a simplicial complex homeomorphic to the $3$-sphere. Let $L$ be a subcomplex homeomorphic to a manifold with nonempty boundary. Show that $H_1(L,\mathbb Z)$ has no torsion. I am totally lost as to how to…
Asvin
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