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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Prove homorphism is an isomorphism

Let $f:X\to Y$ be a map of path connected spaces such that for all $n>0$ the homorphism $f_* :H_n(X;G)\to H_n(Y;G)$ is an isomorphism for $G=\mathbb{Q}$ and for $G=Z_p$ for any prime number $p$. show that the homorphism $f_* :H_n(X;\mathbb{Z})\to…
noname1014
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Is this approach to algebraic topology pursued anywhere?

If we're trying to understand the holes in a space $X$, I guess one way to proceed would be to associate to $X$ a lax functor $\tilde{X} : \mathbb{N} \rightarrow \mathbf{Rel}$ defined as follows. Firstly, if $n \in \mathbb{N}$, then $\tilde{X}(n)$…
goblin GONE
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Invariance of domain for manifold

This is corollary $2B.4$ in Hatcher's Algebraic Topology , state as following : If $M$ is a compact $n-$ manifold and $N$ is a connected $n-$ manifold then an embedding $h : M \to N$ must be surjective , hence a homeomorphism . The first , $h(M)$…
Gankedbymom
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Understanding Switzer's telescope construction on a CW complex

It is my understanding that, given some filtration $(X^n)_{n \geq 0}$ of a space $X$, or more generally a sequence $X^0 \to X^1 \to X^2 \to \cdots$ of spaces and maps, one usually constructs the "telescope" of the sequence as the subspace…
Alex Provost
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Finding a CW complex structure on a quotient space, and then its universal cover

I am given this quotient of the square: I want to: Find a homeomorphic CW complex, Find its fundamental group, Find a simply connected covering space I have seen methods for other quotients of the square like the torus, and $\mathbb{R}P^2$ (for a…
chilliBeanDream
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CW complexes: map on homology induced by inclusions of $n$-skeletons

In discussing CW-complexes, how would I go about proving the following statement: The homomorphism $i_* : H_q(X) \to H_q(X^*)$ is an isomorphism except possibly for $q = n$ and $q = n-1$. Where $X^*$ is a space obtained by 'gluing on' $n$-cells…
QCD_IS_GOOD
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Intuition for first Stiefel-Whitney class

In Hatcher's Vector Bundles and K-theory the following description of orientability of a vector bundle $E \mapsto B$ is given: For a vector bundle $E\mapsto B$ with $B$ path-connected, orientability is detected by the homomorphism $ \pi_1(B)…
user7090
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Index of $p_*(\pi_1(\widetilde{X},\widetilde{x_0}))$ is equal to the number of sheets of the covering

Let $p\colon(\widetilde{X},\widetilde{x_0})\to(X,x_0)$ be a covering space, where both $\widetilde{X}$ and $X$ are path-connected. Note that $p$ is not assumed to be surjective. In Prop. 1.32, Hatcher establishes a bijection between the right-cosets…
user363520
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A sphere covering a space with fundamental group Z/5Z

I am to find a space whose universal cover is a sphere and its fundamental group is Z/5Z. Does anybody have an idea how to approach the problem?
Halinka
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Retract and homology

If $ A\subset X$ is retract of $X$ and the inclusion $i :A\to X$ is a cofibration, prove that: $$\widetilde{H}_n(X)\equiv \widetilde{H}_n(A)\bigoplus\widetilde{H}_n(X/A).$$ where $X/A$ is quotient space, and $\widetilde{H}_n$ is the reduced homology…
noname1014
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Universal cover of wedge sum $T^2 ∨ T^2$

I am trying to construct the universal cover of a wedge sum of two tori. My question is: Can I treat $T^2 ∨ T^2$ as a special case of a two-holed torus with an octagon as its representation? Would appreciate any hints. Thanks
Halinka
  • 144
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fundamental group of two torus

Let $Y_1 = S^1 \times D^2$, $Y_2 = S^1 \times D^2$ and $T_1,T_2 = S^1 \times S^1$. We consider $X=Y_1\cup Y_2 / $~ where ~ is $x \in T_1$ ~ $x \in T_2$. I would like to find the fundamental group of X. I see $X$ as two solid torus which have been…
Sarah
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Fundamental group of $\mathbb{R}^2 \setminus (\mathbb{Z} \times \{0\})$

Good evening, I need to compute $\pi_1(\mathbb{R}^2 \setminus (\mathbb{Z} \times \{0\}))$. I see that the set $\mathbb{R}^2 \setminus \mathbb{Z} \times \{0\}$ has a countable set of points less than whole $\mathbb{R}^2$, so at each point we can…
Diego Vargas
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Relative homology of quotient spaces

Doing some exercises i often encounter arguments like $$H_0(S^2,A)=H_0(S^2/A, A/A).$$ Here $A$ is finite collection of points and $S^2$ is the two-dimensional sphere. Why does this hold? It seems, that this is a general fact, so I would be…
klirk
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Degree of a map $f * g \colon S^p * S^q \to S^p * S^q$

Let $f \colon S^p \to S^p$ and $g \colon S^q \to S^q$ be continuous maps and consider a map $f*g \colon S^p * S^q \to S^p * S^q$ given by $(1-t)x+ty \mapsto (1-t)f(x) + tg(y)$. Here $*$ denotes join. Since $S^p * S^q = S^{p+q+1}$, it makes sense to…
user3158840
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