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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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Generator of Homology of $n$-ball relative to the $(n-1)$-sphere

The claim I want to prove is that: Let $n \geq 1$. $H_n(B^n,\mathbb{S}^{n-1})$ is generated by $[\iota_n+S_n(\mathbb{S}^{n-1})]$. Here is my work: Since $\Delta^n \cong B^n$ and $\partial\Delta^n \cong \mathbb{S}^{n-1}$, we identify them. $\bullet$…
Loafy Loafer
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Algebraic topology question (Qual)

I am having trouble with a QR problem. I would appreciate some help. Construct a connected $CW$-complex $X$ with $H_0(X, \mathbb{Z}) = \mathbb{Z}, H_1(X, \mathbb{Z}) = \mathbb{Z}\times \mathbb{Z}/10\mathbb{Z}$ and $H_2(X, \mathbb{Z}) =…
dinky
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Degree of deck transformations $S^n \rightarrow S^n$

I'd like some verification for the following two claims here on page four (b) and (c). That is, if we first suppose $S^{2n}$ covers $X$, then if $d:S^{2n} \rightarrow S^{2n}$ is a nontrivial deck transformation we know that $deg(d) = (-1)^{2n+1}$.…
user9781778
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Relation between the real line and circle

In M Nakahara's Geomertry, Topology and Physics, he states (example 2.4) that the the quotient space(set of equivalence classes) of the real line under the equivalence relation: $x$ is related to $y$ if $x-y=2n\pi$, $n$ is an integer. is the circle…
Mani Jha
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Fundamental group of a 3-dimensional figure

Let $X$ be a regular hexagon in $\Bbb C$ with centre in $\mathbf 0$, and name his sides $a$, $b$, $c$, $d$, $e$ and $f$ counterclockwise. Let's consider the equivalence relation $\sim$, that identifies $a$ with $b$, $c$ with $d$ and $e$ with $f$.…
Dr. Scotti
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Simple cellular homology computation

Here's a very simple cellular homology computation that I'm a little confused about. Put a CW structure on the closed disc $X=D^{2}$ with two zero-cells $v_{0},v_{1}$, two one-cells $e_{0},e_{1}$ connecting the zero cells, and one two-cell $f$ on…
LCL
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Boundary operator in algebraic topology

I am beginning to learn algebraic topology. For a 2-simplex $\{v_0v_1v_2\}$, the boundary is the chain 1-simplexes: $\partial(v_0v_1v_2)=v_0v_1+v_1v_2+v_2v_0$, which naively and intuitively, I see as the triangle that bounds the 2-simplex, that…
Frank
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$\Delta-$complex structure of a space obtained by identifying the boundary of an annulus with twisting

Let $A = [0,1] \times \mathbb{S}^1$ be the standard annulus and let $X$ be the space obtained from $A$ by identifying $\{0\} \times \mathbb{S}^1$ and $\{1\} \times \mathbb{S}^1$ by a map which represents twice the generator of $\pi_1(\mathbb{S}^1)$.…
24601
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How to apply the matrix of a boundary operator on a k-chain

It is said that the boundary operator $\partial_k$ maps a $k$-chain to a $(k-1)$-chain. I've also seen that this operator can be represented with a matrix of dimension $|K^{k-1}|\times|K^k|$. I can't figure out with a simple example how I can…
Rodolphe Vaillant
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Finding a counter example of the Whitehead theorem

Let $X$ be a connected (based) CW-complex, $X$ is called simple if $\pi_1(X)$ acts trivially on $\pi_n(X)$ for all $n\geq 1$. The Whitehead theorem states that a self-map of a simple connected CW-complex $X$ is a homotopy equivalence if and only…
LipCaty
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surjective map $S^n \rightarrow S^n$ of degree zero

Construct a surjective map $S^n \rightarrow S^n$ of degree zero for ah $n\ge 1$. I’ve been struggling with this exercise from hatcher. I know that if the map is not surjective then the degree is zero, but I have no idea how to approach this one.
topology master
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Euler characteristic of $\mathbb{R}$ and $\mathbb{R}^2$?

I´m trying to understand the value of the Euler characteristic of the real line and the real plane. I don´t know if it is defined, I think that it is for any topological space. So this could be right? If we separate $\mathbb{R} = (-\infty,x_0] \cup…
user183002
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Homotopy Type of a Riemann Surface with and without Points Removed

Suppose $\Sigma$ is a Riemann surface of genus $g$ and with $b$ points removed. Is there any restriction on the possible homotopy type that $\Sigma$ can possess? What about the case when $\Sigma$ has no points removed?
user02138
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on the factorization of maps between connected CW complexes

I'm working on problem 16 in section 4.1 of Hatcher's Algebraic Topology book. I really have no ideas so far: Show that a map $f: X \to Y$ between connected CW complexes factors as a composition $X \to Z_n \to Y$ where the first map induces…
Tony B
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Is the induced homomorphism of an onto map necessarily onto?

If $g:X\rightarrow Y$ is an onto continuous map, is the induced homomorphism $g_*:\pi_1(X,x)\rightarrow \pi_1(Y,y)$ onto? Also, does it matter whether $X,Y$ are path-connected or not?
user500144
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