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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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How to prove that a simplicial complex is path-connected if connected?

If K consists of finite simplices and connected, it seems intuitively clear that any two 0-simplices can be connected by a path whose image is a collection of 0-simplices and 1-simplices. But I can't rigorously construct such a path... Could anyone…
Keith
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Covering space of the wedge of the unit circle and the real projective plane

Let $Z * Z/2Z = \langle a, b | b^2=1\rangle$ be represented by $X = S^1\vee RP^2$ i.e. the wedge of the unit circle and the real projective plane. Let $H$ be the smallest normal subgroup containing $b$. Question: How can we construct a covering…
yaa09d
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Proof explanation Mayer-Vietoris and the Punctured Euclidean Space.

I am only releasing part of the proof. Doesn't this prove that "otherwise" case is completely wrong? I am assuming "otherwise" refers to $n < 2$? Because he just showed that $H^0(\Bbb R - \{ p \}) \simeq \Bbb R^2 $, not isomorphic to $0$. I…
Lemon
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Singular homology of cofinite topology space

Suppose $X$ is an infinite set equipped with cofinite topology, what are its singular homology groups? For example, $X=\mathbf{CP}^1$ equipped with Zariski topology (the cofinite topology), what are its singular homology groups? It is connected, so…
user93417
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A question about the proof of general lifting theorem in Munkres topology p.479

after the lifting $\tilde f$ of $f$ is defined, the book goes on to prove its continuity. And in that part, the book says that replacing $U$ by a smaller neighborhood, $V_0$ can be contained in $N$. But why is it possible? How to construct a smaller…
Keith
  • 7,673
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Computing the homology of a triangle with the edges identified in cyclic order

Can someone help me with computing the simplicial homology of the following space, and also clarify why it is a delta complex in the first place: (The picture means that the edges are identified in the given orientation, also please note that the…
Hajime_Saito
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Conceptualizing Inclusion Map from Figure Eight to Torus

I'm having some difficulty getting an understanding of this issue: I have an inclusion map $i : S^1 \vee S^1 \hookrightarrow S^1 \times S^1$. So this is an inclusion map from the figure eight to the torus. If the leftmost circle of the figure…
user21787
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Hatcher's formula in homotopy equivalence proof

In the proof that two homotopic maps induce the same homomorphism in homology, appears the formula (bottom of p. 112, Hatcher, Algebraic Topology): \begin{gather} P(\partial \sigma) = \sum_{i
Aldebaran
  • 125
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Homomorphisms induced by maps $S^1 \times S^1 \rightarrow S^1 \times S^1$.

Problem 2.2.30 in Hatcher involves the homomorphisms $H_2(S^1\times S^1) \cong \mathbb{Z} \rightarrow H_2(S^1\times S^1) \cong \mathbb{Z}$ induced by The map $S^1 \times S^1 \rightarrow S^1 \times S^1$ that is the identity on one factor and a…
user109360
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Homotopic paths implies equal winding numbers

I am trying to prove a proposition relating analysis and geometry. I have a general idea on how to prove it. However, a small part of the proof needs a lemma about path homotopy and winding number. Specifically, here it is: Let…
Haley13
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boundary in homology group

f is a reflection on a sphere $S^{n}$, $\sigma_{1}$ is a diffeomorphism from $D^{n}\subset \mathbb{R}^{n}$ to one of the two caps of the sphere, separated by the plane of the reflection and $\sigma_{2}=f(\sigma_{1})$. How can I show that…
user166147
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Quotient map not nullhomotopic

I have the following qual problem: Let $M$ be a connected closed surface, not necessarily orientable, with an embedded closed disk $D$. Let $Q$ be the quotient space of $M$ by $\overline{M\setminus D}$. After showing $Q\cong S^2$ show that the…
tghyde
  • 2,055
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Homology of a finite graph follows from Mayer-Vietoris sequence?

Problem (Fulton's Algebraic Topology: A First Course, Exercise 10.15) If $X$ is a finite graph with $v$ vertices and $e$ edges, and $X$ has $k$ connected components, show that $H_1X$ is a free abelian group with $e-v+k$ generators. Thoughts First,…
Yai0Phah
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What's the meaning of this about the cyclic fundamental group?

"You can also get a cyclic group of order p by attaching a disk to a circle by wrapping it around the circle p times (the fact that the fundamental group is Z/pZ follows from Van-Kampen’s theorem). " But I can't understand what's the figure looks…
6666
  • 3,687
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Rank of $H_1(X)$ for a retract of $M_g$

I'm trying to solve the following exercise and I would be grateful if I could get a hint: If the closed orientable surface $M_g$ of genus $g$ retracts onto a graph $X\subset M_g$, then the rank of $H_1(X)$ must be at most $g$. Since there is a…
user54631
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