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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 the mod $\mathfrak{C}$ Whitehead theorem from the mod $\mathfrak{C}$ Hurewicz theorem

I am reading Lecture Notes in Algebraic Topology by Davis and Kirk and in the book the mod $\mathfrak{C}$ Whitehead Theorem (where $\mathfrak{C}$ is a Serre class of abelian groups) is stated as: mod $\mathfrak{C}$ Whitehead Theorem: Let $f : A \to…
Perturbative
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Retraction of a Moebius strip

Let $X=([0,1]×[0,1])/\sim$ , where $\sim$ is the equivalence relation generated by $(0,y)\sim (1,1-y)$, and let $Y=([0,1]×\{ 0,1 \} )/\sim \ \subset X$. I must show that there are no retractions $r:X\to Y$. First off, $X$ is homeomorphic to a…
Dr. Scotti
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Exactness of Mayer Vietoris Sequence

I am confused in showing that the sequence is exact at $A_{n-1}' \oplus B_{n-1}$. Here is part of my argument. Note that $(\rho_4+f_5)\circ (f_4,-\delta_4)=\rho_4 \circ f_4-f_5 \circ \delta_4=0$ by commutativity of the diagram. This shows that…
Loafy Loafer
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Prove that $[0,\infty) \times \mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^3$

Prove that $[0,\infty) \times \mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^3$. I want to do this using the tools of algebraic topology. Some thoughts I have on the matter is that if we remove a point from $\mathbb{R}^3$ then it becomes…
user637978
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Special case of invariance of domain

Let $A=\{(x_1, \cdots, x_n)\in \mathbb{R}^n: x_1\ge 0,\|(x_1, \cdots, x_n)\|<1\}$. I want to show that this is not homeomorphic to any open set of $\mathbb{R}^n$. I can use the theorem of invariance of domain, which states that: If $U$ is open in…
Spook
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The calculation of a homology group

This is a question on Massey's A Basic Course of Algebraic Topology. I met some problem in calculating a homology group of a specific space when dealing with a question. Let $X=\{ (x,y,z) \mid xyz=0 \}$ then how to calculate $H_2(X)$? By the way,…
Dick. Y
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Computing homology groups of two mobius strips glued together by four-fold covering map

Let $M_1$ and $M_2$ be two mobius strips with boundaries $S_1$ and $S_2$, respectively. Say we form a space $X$ by gluing $M_1$ and $M_2$ along their boundaries by a 4-fold covering map, $f: S_1 \rightarrow S_2$. Compute $H_k(X)$. So, $H_2(M_i)=0$…
user637978
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Cohomology algebra of a differential graded algebra

I am struggling to understand how to think about cohomology algebra of DGA. In particular, let $H^*$ be cohomology algebra of a differential graded algebra $(A,d)$. I know $(A,d)$ is the cochain complex: $$\dots\longrightarrow…
Aleksandar
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Homology of $n$-dimensional Torus

This webpage describes the homology groups $H_k(T^n)$ of the $n$-dimensional torus as being $\mathbb{Z}^{{n}\choose{k}}$. There is a very helpful example in Hatcher, example 2.39, where he works through the case of $n=3$. It isn't obvious to me,…
Jim
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Show that two embeddings of $M$ into its product are not homotopic

Assume that $M$ is a compact smooth manifold with positive dimension. We have two ways of embedding $M$ into its product with itself. Way I: $ i_1(m) = (m, m)$ and Way II: $i_2(m) = (a, m)$, where $a \in M$. Show that those two maps cannot be…
koch
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Fundamental group of a solid torus with a tunnel?

Imagine that you have a donut and a worm inside. This worm takes two turns around the solid torus, going back to the starting point after two laps. How could I find out what the fundamental group of this space is? Can I use Van-Kampen's theorem to…
josmat
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$\pi_1(U\cap V)\cong\mathbb{Z}$ for $U,V$ simply connected

Let $X=U\cup V$ where $U,V$ are simply-connected open sets and $U\cap V$ is the disjoint union of two simply connected sets. We also have the condition that any subspace $S$ of $X$ homeomorphic to $[0,1]$ has an open neighborhood that deformation…
user500144
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What is the standard homotopy equivalence?

The capture is from the book "R.M.Switzer, Algebraic Topology-Homology and Homotopy". I want to know the homotopy type of the so-called "standard" homotopy equivalence $\mu:S^m\wedge S^n\longrightarrow S^{m+n}$, where $S^m:=S^{m-1}\wedge S^1$. Let…
LipCaty
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Homology functor into an abelian category instead of abelian groups.

All of the Eilenberg-Steenrod axioms for homology can relatively easily be translated into the language of category theory. We can then replace the abelian groups with a general abelian category. Has this been done before? If not what complications…
Noel Lundström
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Isomorphisms of CW Coverings

This is an exercise I've been working on from Hatcher (1.3.32): Consider covering spaces $p: \tilde{X} \to X$ with $\tilde{X}$ and $X$ connected CW complexes, the cells of $\tilde{X}$ projecting homeomorphically onto $X$ cells. Restricting $p$ to…
ec92
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