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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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How to show that Postnikov towers are fibrations?

I cannnot understand why postnikov towers are fibrations. For example, consider a postnikov tower for 1-dimensional shpere $S^1$. Then, Let $X^0$ be a disc whose boundary is the sphere $S^1$. Let $X^1 = S^1$. Then I guess, the inclusion map $p:X^1…
Hausdorff
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What is the topological space with fundamental group isomorphic to this presentation $\langle a,b \mid a^2=b^2\rangle$?

In this answer, it talks about how you can create a topological space with the following fundamental group: $$\langle a,b \mid a^2=b^2\rangle$$ The answer does this by creating a topological space consisting of two unit circles that are connected to…
abetray
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The Lefschetz Fixed Point Theorem in Coefficient Ring $G$

As you know Lefschetz number is $\tau (f)\doteq \sum_n (-1)^n {\rm tr}(f_\ast : H_n(X)\rightarrow H_n(X))$ where $X$ is a finite simplicial complex and $f : X\rightarrow X$. So LFPT is that if $\tau(f)\neq \emptyset$, then $f$ has a fixed point.…
HK Lee
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Fixed point property on the torus

Consider the torus $T = \Bbb S^1 \times \Bbb S^1$. Show that $T$ does not have the fixed point property. A space $X$ is said to have the fixed point property if for any continuous map $f : X \to X$ there exists $x \in X$ such that $f(x) = x$. I…
Walker
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Hatcher Exercise 2.2.30(a)

For the mapping torus $T_f$ of a map $f:X\to X$, we have a long exact sequence $$\cdots\to H_n(X)\xrightarrow{1-f_*}H_n(X)\to H_n(T_f)\to H_{n-1}(X)\to\cdots.$$ Use this to compute the homology of the mapping tori of the following maps: (a) A…
one potato two potato
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Surgery diagram and fundamental group

Let $Y^3$ be a 3-manifold obtained by surgery on $S^3$ along hopf-link with framing $p,q\in \mathbb{Z}$. I know that $Y^3\cong L(pq-1,p)$ from the Rolfsen twist. But, I wonder how can I compute the fundamental group from the surgery diagram…
John Smith
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Find all the covering spaces of the Klein Bottle

I am studying for a qualifying exam and it asks for all the covering spaces of the Klein Bottle. Does anyone have any suggestions? I am not sure how to go about finding ALL of them.
dunkindonuts
  • 143
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eilenberg-steenrod for pairs in any model category?

The Eilenberg Steenrod axioms are functors on the homotopy category of pairs of "spaces" $(X, A)$. Typically they are introduced when $X$ and $A$ are some sort of topological spaces. My question is can "spaces" be taken to mean objects in other…
ykm
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Questions regarding homotopy of paths vs homotopy of loops

Given a space $X$ and a path-connected subspace $A$ containing the basepoint $x_0$, show that the map $\pi_1(A,x_0) \to \pi_1(X,x_0)$ induced by the inclusion $A \to X$ is surjective iff every path in $X$ with endpoints in $A$ is homotopic to a…
Emil Grönberg
  • 277
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Are countable CW complexes closed under loop spaces?

It’s well know that finite CW complexes are not closed under loop spaces (viewed as homotopy pullbacks). Just consider the circle. Are countable-dimensional CW complexes with countably many cells at each level closed under these homotopy pullbacks?…
CuriousKid7
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The homotopy class of a path in I x I?

I am beginning study in algebraic topology and came across this problem, which is rather upsetting to me. My understanding is that elements of the fundamental group are loops, so how is this dashed path defined? If it is meant to represent the loop…
user799930
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2 answers

Definition of map to barycentric subdivision?

I'm reading the Rotman's An Introduction to Algebraic Topology, p.247 and I'm confused with some notation. What is the $\operatorname{Sd} : K \to \operatorname{Sd} K$ ? Here, $\operatorname{Sd}K$ is the barycentric subdivision. Note that…
Plantation
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Minimal number of contractible sets covering $\mathbb{CP}^3$

In an exam recently, I was asked to find the minimal number of contractible sets covering $\mathbb{CP}^3$ by considering the cup-product on relative cohomology. Is there nice a way of doing this, either using the proposed approach or some…
Raeder
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Help calculating the second homology group of $\mathbb{R}P^2 \times S^1$

I need help calculating the second homology group of $\mathbb{R}P^2 \times S^1$. I found all the other homology groups using the Mayer-Vietoris sequence. Any suggestions? I can't use Kunneth.
dunkindonuts
  • 141
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1 answer

Homology of a boundary in ambient manifold

Let $M$ be a compact oriented $m$-dimensional manifold with a boundary $N \neq \emptyset$. Then $N$ is a closed oriented $m-1$-dimensional manifold. Is it always true that the homology class $[N]_M$ has to vanish in $H_{m-1}(M, \mathbb Z)$?
Terry Black
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