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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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How to show the reduced cone is homeomorphic to $D^2$?

The reduced cone is given by: $$CX = （X \times I） /(X \times 1 ∪ x_0 × I),$$ where $I$ is the unit interval $[0,1], x_0$ is the base point. Let $x_0$ be any point in $S^1$. How can I show that $CS^1$ is homeomorphic to $D^2$? $S^1$ is the unit…
Hui
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Prove that a space is simply connected

So I've been working through this book (A Basic Course in Algebraic Topology, by William Massey) in preparation for an Algebraic Topology course I'm going to take soon, and I ran into trouble with this exercise. Let ${U_i}$ be an open covering of…
mynameissurge
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Algebraic Structure of the Rose with Two Petals

I am trying to determine whether the rose with two petals ($S^1 \vee S^1$ or the figure-eight) has a continuous multiplication with identity element. I know that this is true for the unit circle $S^1$ in the complex plane, where $S^1 = \{ z \in…
rfxoneill
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A basic question on homology

Let $X$ be Hausdorff. Suppose further that it is triangulable. Let $K$ and $L$ be two simplicial complexes such that their underlying space $|K|=|L|=X$. It is a lot of work to show (see Chapter 2 of Munkres' Elements of Algebraic Topology) that the…
doofus
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$H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$ isomorphism in algebraic topology

In algebraic topology we have the result $$H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}.$$ In Massey's book, this is a result that follows from the fact that the sequence $$0 \rightarrow \tilde{H}_0(X) \xrightarrow{\xi} H_0(X) \xrightarrow{\xi_*}…
Numbersandsoon
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Euler characteristic of a space minus a point

Let $X$ be a topological space and $*$ be the base point of $X$. How does $\chi(X-*)$ relate to $\chi(X)$ do we have $\chi(X-*)=\chi(X)-\chi(*)=\chi(X)-1$?
palio
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Composed Covers

I have problems solving this seemingly straightforward question. Let $q : X \rightarrow Z$ be a covering space. Let $p : X \rightarrow Y$ be a covering space. Suppose there is a map $r : Y \rightarrow Z$ such that $q = r \circ p$. Show that $r : Y…
Jake
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Discrete Subgroup is Regular Covering Space

I'm lacking ideas how to attack the following problem: Given $G$ a topological group, and $H$ a discrete subgroup, I have to show that that $G \rightarrow G/H$ is a regular covering space. Could someone give me a pointer? Thank you Stefan
stefan
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Some questions about homology with local coefficients.

If $F:\pi_1(X)\rightarrow Ab$ is a system of local coefficiens, on the topological space $X$, then we can define the homology of $X$ with coefficients $F$ by taking the homology of the chain complex $C_p(X,F)=\bigoplus F(\sigma(e_1))$ where the sum…
Bill
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Null-homotopic covering space map

I'm stuck with the following question, which looks quite innocent. I'd like to show that if a covering space map $f:\tilde{X}\to X$ between cell complexes is null-homotopic, then the covering space $\tilde{X}$ must be contractible. Since $f$ is…
user54631
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Extending cellular maps between aspherical complexes

In a paper I read, the author seemed to use a property similar to: Let $X, Y$ be two aspherical CW-complexes and $f : X^{(2)} \to Y^{(2)}$ be a cellular map between their 2-skeletons. Then $f$ extends to a cellular map $\tilde{f} : X \to…
Seirios
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Algebraic question (In Hatcher's book, exercise: 1.1.16-c:)

In Hatcher's book, exercise: 1.1.16-c: Show that there are no retractions $r :X \rightarrow A$ in the following cases: (c) $X = S_1 × D_2$ and A the circle shown in the figure. Page 39 By the inclusion, the induced map from the fundamental group…
6666
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Cohomology of $\mathbb RP^{n}$ with $\mathbb Z_2$ coefficients

In http://en.wikipedia.org/wiki/Universal_coefficient_theorem they use the Universtal coefficient theorem $$0 \rightarrow \mbox{Ext}(H_{i-1}(X, \mathbb{Z}),A)\rightarrow H^i(X,A)\rightarrow\mbox{Hom}(H_i(X, \mathbb{Z}),A)\rightarrow 0$$ to…
palio
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Hatcher 1.3.25: finding $\pi_1$ of the quotient of the punctured place by a hyperbolic $\mathbb{Z}$-action

Problem 1.3.25 in Hatcher's "Algebraic Topology" is concerned with the quotient (call it $Y$) of $X=\mathbb{R}^2 \setminus \{0\}$ under the $\mathbb{Z}$-action generated by $(x,y) \mapsto (2x,y/2)$. Some parts of the problem are obvious to me ($Y$…
bob
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$\mathbb{R} \mathbb{P}^2$ as the total space of a covering

Suppose there exists a covering $\xi :\mathbb{R} \mathbb{P}^2 \rightarrow X$. How can I show that $\xi$ is a homeomorphism? Thanks!
jules
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