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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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Are all maps into path-connected spaces homotopic?

Context: A problem I'm solving asks "Prove that any two maps $S^m \rightarrow S^n$, where $n>m$ is homotopic. [Hint: use the Simplicial Approximation theorem] My first thoughts were that if we have two maps $f,g$, since $S^n$ is path connected, I…
Solveit
  • 838
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1 answer

Isomorphism between homology group and reduced homology group of mapping cone

Given a map $f : X \to Y$, the mapping cone $C(f)$ is the space obtained from the mapping cylinder $M(f)$ by identifying the subspace $X \times \{0\}$ to a single point. How can I construct an isomorphism between the homology group $H_n(M(f),X…
Danny
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3-manifolds with certain fundamental group

Suppose $M$ is a 3-manifold with $\pi_1(M)=\mathbb{Z}/p\mathbb{Z}$. Is it true that $M$ is diffeomorphic to a lens space $L(p,q)$?
7779052
  • 963
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Mapping $f: (S^n, pt) \to (S^n, pt)$ induces multiplication by $deg(f)$ on $H_*(S^n \wedge X, pt)$.

I'm studying for my final exam in Algebraic Topology and got stuck with somewhat straightforward looking preparation problem. Let $\tilde H$ be any reduced homology theory and $f: (S^n, pt) \to (S^n, pt)$ continuous mapping. Show that for every…
Adam Baranowski
  • 545
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Calculating the fundamental group of the Klein Bottle using the Seifert-Van Kampen theorem

I want to calculate by two different way the fundamental group of the Klein Bottle. First one: I want to use that the Klein Bottle is can be decomposed in two Mobiüs Band as the following picture shows: Then I have to choose two open set $U$ and…
EQJ
  • 4,369
4
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4 answers

Relationship between homology of suspension of $X$ and $X$

The exercise is the following: Show that, for any homology theory (satisfying the usual axioms), there is a natural isomorphism $ \tilde{H_i}(X) \rightarrow \tilde{H}_{i+1}(\Sigma X)$. Well, I tried using the long exact sequence: $$...\rightarrow…
Aloizio Macedo
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4
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How to prove $\deg( f\circ g) = \deg(f) \deg(g) $?

If $ f,g:S^1 \rightarrow S^1$ continuous maps then \begin{equation*} \deg( f\circ g)= \deg(f)\deg(g). \end{equation*} Unfortunately, i haven't made any progress in solving it. I've tried considering the lift of f,g and $f\circ g$ but i can't see…
Mariospkt Pkt
  • 131
4
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Winding maps of spheres?

I know that the second homotopy group of $S^2$ is $\mathbb{Z}$. I also know that a simple representative of the class of maps that generates this group is the identity map on the sphere, $i:S^2\rightarrow S^2, p \rightarrow p$ In other words, this…
Tom
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Covering Map of Torus

how can I show that the following map is a covering map of $T:=$ $S^1$ x $S^1$? $\pi: T\rightarrow T$ with $(x,y)$ $\mapsto$ $(x^ay^b, x^cy^d)$, where $a,b,c,d \in \mathbb{Z}$ and $ad-bc=m\neq 0$. Furthermore every $(x,y)$ has $|m|$ inverse images…
Alex
  • 51
4
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1 answer

Small doubt about the connecting homomorphism on the long homology sequence

When you consider the long homology sequence (of spaces $A,X$ , with $A$ subspace of $X$) you need to define an homomorphism from $H_q(X,A)$ to $H_{q-1}(A)$ to obtain the long homology sequence from the short one involving $0 \to H_q(A) \to H_q(X)…
DCao
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Computing a fundamental group and describing a universal cover

This is the second half of exercise 1.3.21 from Hatcher. Let $Y$ be the space obtained by attaching a Möbius band $M$ to $\mathbb{R}P^2$ via a homeomorphism from its boundary circle to a circle in $\mathbb{R}P^2$ lifting to the equator in the…
Bey
  • 3,893
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1 answer

References request on characteristic class

I am planing to learn something about characteristic classes on my own. I am wondering if anyone could recommend me something on such materials like constructions of vector bundles, Thom isomorphism theorem with applications and intersection theory.…
Jason785
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4
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What is on the cover of Hatcher's Algebraic Topology book?

What is on the cover of the book? Is it the Hopf fibration?
bob villa
  • 49
4
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2 answers

Mapping cylinder cofibration

Let $f:X\to Y$ be a continuous map, and let $M_f = (X\times I) \sqcup Y)/(x,0)\sim f(x)$ be its mapping cylinder. Then the inclusion $X\to M_f$ is a cofibration. My attempt: Using the following theorem from Bredon seems like the most promising…
iwriteonbananas
  • 6,102
4
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1 answer

Inclusion is cofibration iff $(A\times I) \cup (X\times \{0\})$ is a retract of $X\times I$

Let $A\subset X$ be a subspace. Then the inclusion $i:A\to X$ is a cofibration iff $(A\times I) \cup (X\times \{0\})$ is a retract of $X\times I$. I've proved the "$\implies$" direction. “$\impliedby$“: Suppose that $H: X\times I \to Y = (A\times…
iwriteonbananas
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