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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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Embeddability of the cone of Klein bottle to $\mathbb R^4$

By using Alexander Duality, we can show that $K\not\hookrightarrow \mathbb R^3$ and we can also give an explicit formula for $K\hookrightarrow\mathbb R^4$ (here $K$ is the Klein bottle). How about the cone $CK$? Can it be embedded in $\mathbb…
Y.H. Chan
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Zeroth homotopy group: what exactly is it?

What are the elements in the zeroth homotopy group? Also, why does $\pi_0(X)=0$ imply that the space is path-connected? Thanks for the help. I find that zeroth homotopy groups are rarely discussed in literature, hence having some trouble…
yoyostein
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Question about the first proof in Hatcher's Algebraic Topology

I have a question about Hatcher's proof that the fundamental group of a circle is Z. Specifically, halfway through, ( http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf , page 30), he proves an important a lemma stated rather generally: Given a map…
Lost In Math
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Covering spaces Hatcher question 6.

Let X be the shrinking wedge of circles. Which is the radius of circles $X \in R^2$ such that it's the union of $C_n$ circles centered at $(\frac{1}{n},0)$ with radius $\frac{1}{n}$ for n=1,2,3... I'm having a bit of trouble with sheeted aspect.…
simplicity
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Constant maps induce zero homomorphism

It seems reasonable for me that if $f:X\rightarrow Y$ is the constant map then $f_{*}:H_{n}(X)\rightarrow H_{n}(Y)$ is the zero map for $n>0$. But I don't see how to prove this. If $n$ is odd then it is OK as if $[p]_{n}$ denote the class of unique…
Bingo
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Cohomology of wedge equals direct sum of cohomologies

I have seen the following fact used somewhere (for example to show that $\mathbb{RP}^3$ is not homotopy equivalent to $S^3\vee\mathbb{RP}^2$): Let $X,Y$ be two path connected pointed spaces such that the base points each have a contractible…
Daniel Robert-Nicoud
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Surface of genus $g$ does not retract to circle (Hatcher exercise)

I'm trying exercise 9 on page 53 in Hatcher but I need some help with it. The exercise is: In the surface $M_g$ of genus $g$, let $C$ be a circle that separates $M_g$ into two compact subsurfaces $M_h^\prime $ and $M_k^\prime$ obtained from the…
Rudy the Reindeer
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Homology of quotient of 3-sphere by identifying antipodal points on equator

I'm trying to solve exercise 2.2.10 in Hatcher's Algebraic Topology: Let $X$ be the quotient space of $S^{3}$ under the identifications $x\sim-x$ for $x$ in equator $S^2$. Compute the homology groups $H_i(X)$. $H_i(X) = 0$ for $i > 3$.…
PeterM
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Proof: A loop is null homotopic iff it can be extended to a function of the disk

I would like to prove the following basic fact related to the fundamental group: A loop $\gamma : [0,1] \rightarrow X$ is null homotopic if and only if it can be extended to a continuous function of the disk $D^2$. Can you tell me if this is…
Rudy the Reindeer
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Question on the symmetric product $\mathrm{Sym}^g\Sigma$

Let $\Sigma$ be a genus $g$ closed Riemann surface. Recall that the symmetric product $\mathrm{Sym}^g\Sigma$ is the quotient of the $g$-fold product $\Sigma \times \dots \times \Sigma$ under the action of the symmetric group $\mathfrak{S}_g$ on $g$…
jkr
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Proving that the cohomology ring of $\mathbb{R}P^n$ is isomorphic to $\mathbb{Z}_{2}[x]/(x)^{n+1}$

We have to prove that $H^{*}(\mathbb{R}P^n, \mathbb{Z}_2) \simeq \mathbb{Z}_{2}[x]/(x)^{n+1}$ as ring. So we have to find an isomorphism $$ \phi: \mathbb{Z}_{2}[x]/(x)^{n+1} \rightarrow H^{*}(\mathbb{R}P^n, \mathbb{Z}_2) $$ We can send $x$ in a…
ArthurStuart
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Equivalence of two definitions of Whitehead Torsion

In their book Lecture Notes in Algebraic Topology, Davis and Kirk define the torsion of an acyclic chain complex $C$ in the following way: Since $C$ is acyclic, there exists a simple chain complexes $E,F$ and a chain isomorphism $f: E \rightarrow F…
user39598
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Show a subspace formed by a Klein bottle is homotopy-equivalent to $S^1 \vee S^1 \vee S^2$

I am trying to solve Hatcher, chapter 0, 20: Show that the subspace $X \subset \mathbb{R}^3$ formed by a Klein bottle intersecting itself in a circle, as shown in the figure, is homotopy equivalent to $S^1 \vee S^1 \vee S^2 =…
Dávid Natingga
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Difference between Homology and Cohomology

Homology and cohomology are similar because the latter is the former acted by $\text{hom}$ functor, and we also have Theorem Let $C$ and $D$ be free chain complexes; let $\phi:C\to D$ be a chain map. If $\phi$ induces homology isomorphisms in all…
gaoxinge
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How to show the fundamental group of torus is abelian in a homotopic way?

I know the torus is homeomorphic to $S^1 \times S^1$ and the fundamental group is $ \mathbb{Z} \times \mathbb{Z} $, but in the real case, (let the generators of the torus's fundamental group be $a$ and $b$). Like in the case of $S^1 \vee S^1$…
6666
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