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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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Why is covering map a fibration? Incorrect proof?

This is on page 66, Tom Dieck's A covering $p:E \rightarrow B$ is a fibration. What concerns me is its proof. The proof is claims is suffices to consider local lifts - how? On page 63, (Prop 3.1.3) there is a uniquely lifting from $f:X…
Bryan Shih
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Sign convention for compatibility of cap and cross products

Bredon's Topology and Geometry contains the following formula relating cap and cross products (Theorem 5.4 in chapter VI): $(\alpha \times \beta) \cap (a \times b) = (-1)^{\deg(\beta) \deg(a)} (\alpha \cap a) \times (\beta \cap b)$ for $\alpha \in…
jconder
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triangulation of a compact surface $v \leq f$

For any triangulation of a compact surface, show that $$v \leq f$$ where $v,e,f$ is vertices, edges and triangle respectively. Since each triangle has 3 edges and each edge is the common edge of exactly two triangles, we get $3f=2e$ and $v-e+f=…
Desunkid
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A detail in the proof of Poincaré duality

In Hatcher's Algebraic Topology, on page 246 (here, in the book), about two-thirds down the page he states that showing the commutativity of the two squares shown in the diagram, not involving the boundary or coboundary maps, is a 'triviality'. I…
Hargrove
  • 107
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$X$ is simply connected iff every continuous $f:S^1 \rightarrow X$ has a continuous extension to the unit disc?

Given that a topological space $X$ is path connected, I need to show that $X$ is simply connected iff every continuous $f:S^1 \rightarrow X$ has a continuous extension to the unit disc. My professor explained to me the proof of the $\leftarrow$…
Wyatt Gregory
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Proof of another Hatcher exercise: homotopy equivalence induces bijection (part II)

This is part 2 of the question I asked here. Can you tell me if this proof is correct? I thought it's better to ask this in a new separate question, the old post was getting a bit long. Many thanks for your help! Claim: $f$ restricts to a homotopy…
Rudy the Reindeer
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infinite product of $1$-sphere has no universal covering space

This is example 16 in Chapter 2, Section 5 of Spanier's Algebraic Topology. It states that: Any infinite product of $1$-sphere has no universal covering space. A universal covering space, as far as I know, need not to be simply connected. One…
du.Du.DU
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On embedding of a triangulation

Let $X$ be a triangulable topological space, i.e, a topological space which is homeomorphic with a finite simplicial complex. Suppose $X$ can be embedded in $\mathbb{R}^d$ for some $d$. Is it possible to find a simplicial complex inside…
123...
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Why is $k$ called a "natural morphism" here? (van Kampen's Theorem)

Source: https://en.wikipedia.org/wiki/Seifert%E2%80%93van_Kampen_theorem I am aware of "natural transformations" in category theory. However, I have trouble figuring out in this context why is $k$ called a "natural morphism"? I can't figure out…
yoyostein
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Algebraic Topology-Explanation required for the following definition

I am currently reading the book A combinatorial introduction to topology by Michael Henle. Under "Compactness and Connectedness" there is the following definition which I didn't understand at all. I do know the definition of nearness. Let…
RagingBull
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Is there an argument that generalizes this proof to Bézout's theorem?

Letting $C_1$ and $C_2$ be two algebraic curves sitting in $\mathbb CP^2$, they both realize homology classes in $H_2(\mathbb CP^2)$ and are in fact represented by $n \cdot [\mathbb CP^1]$ and $m \cdot [\mathbb CP^1]$. If the curves intersect…
Andres Mejia
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If $r : X \to A$ is a deformation retract and $i : A \to X$ is inclusion, then $i(r)$ is homotopic to $id$ on $X$

If $r : X \to A$ is a deformation retract and $i : A \to X$ is inclusion, then $i(r)$ is homotopic to $id$ on $X$. I know that since r is a deformation retract, it is homotopic to id on X, but I don't see an obvious construction of the above…
Dan Donnelly
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Show that any map $f: S^{a + b} \to S^a \times S^b$ induces a zero map on the homology $H_{a + b}$

Show that any map $f: S^{a + b} \to S^a \times S^b$ induces a zero map on the homology $H_{a + b}$. I'm trying to prove this statement. My thought is that: Showing this is equivalent to showing that $f$ induces a zero map on the $H^{a+b}$ by the…
nekodesu
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Show that every map $S^n \to S^n$ can be homotoped to have a fixed point if $n > 0$.

This is exercise 2.2.6 on Hatcher. I have seen same question on MSE but I'm having a hard time visualizing the homotopy that many people described. Wondering if someone can explain to me more explicitly and intuitively. For example, here in this…
nekodesu
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What topological space do I obtain by gluing these edges?

I'm gluing the edges of a square together with the caveat that there's a "fold" down the middle. I think this produces sort of a sphere with four "pinches". I'm wondering if my intuition is correct and if someone could provide a more rigorous…
Jackson Walters
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