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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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What is the moduli space of lines in $\mathbb R^3$?

If we restrict to those lines passing through the origin, we of course get $\mathbb{R}P^2$. Is there a good topological description of the space that we get when we remove the restriction that they pass through the origin? Is there a name for this…
Carl
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Minimal Degree of map $S^2\times S^2\mapsto \mathbb{CP}^2$

I am having troubles finding the minimal d such there is a map of such degree from $S^2\times S^2\mapsto \mathbb P^2$. I know that the cohomology ring of $\mathbb P^2$ is thus I know that the degree will be given by $f^*(x^2)=(f^*(x))^2$ which will…
Spotty
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Attaching a 2-cell by a word

In Example 2.36 on Pg 141 of Hatcher's Algebraic Topology, he writes: ... one 2-cell attached by the product of commutators $[a_1,b_1] \ldots$ Can someone please explain to me what is meant by attaching a 2-cell by a word. I am assuming it means…
doofus
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Nontrivial cycles in the zero set of a map

Let $B_4$ be a 4-disc, $X:=S^1\times B_4$, $\partial X=S^1\times S^3$ and $f: X\to\mathbb{R}^3$ be continuous, such that $f|_{\partial X}: (s_1, s_2)\mapsto H(s_2)$, where $H: S^3\to S^2\subseteq\mathbb{R}^3$ is the Hopf fibration. How to show that…
Peter Franek
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Mayer-Vietoris Type Sequence For Pushouts

Pushouts in the category $\mathsf{Top}$ of topological spaces exist and under certain conditions are known as adjunction spaces. Rigorously, if is a diagram in $\mathsf{Top}$, then there exists a universal commutative…
Brian Fitzpatrick
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Is it possible to define orientability using orientation preserving loops?

Wikipedia says that the orientable double cover corresponds to the subgroup of orientation preserving loops in $\pi_1$ (which is of index 1 or 2 apparently). My questions are: What is an orientation preserving loop? Is it possible to use it to…
user40167
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Mapping cylinder of $z\rightarrow z^2$

A question was asked in my topology course the other day (not an assignment for credit). Let $f:S^1\rightarrow S^1$ by $f(z)=z^2$ ($S^1$ is considered to be in the complex plane). What is the mapping cylinder of $f$? After discussing it briefly with…
Fred Byrd
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An h-cobordism problem

Im trying to understand the proof of Lemma 2.3 of Milnor and Kervaire: Groups of homotopy spheres I. Suppose we have a simply connected manifold $M$ which bounds a contractible manifold $W'$. Then the Lemma says, that $M$ is h-cobordant to the…
unknownMathematician
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Leray-Hirsch theorem

I'am studying the book "Bott, Tu Differential forms in algebraic topology." I don't understand the proof of Leray-Hirsch theorem via Cech-de Rham complex. Lets consider some bundle $\pi: E \mapsto M$ with fiber $F$ and some good open cover…
Gleb
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Regular CW complexes are homeomorphic to delta complexes

I am trying to read Algebraic topology book by Hatcher. On page 535 he has stated that regular CW complexes are homeomorphic to $\Delta$-complexes. My question is that why is this true? Just an easy visible example is helpful.
Pouya Layeghi
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Hatcher 3.3 Exercise 31

The following is a question from Hatcher's "Algebraic Topology": Let $M$ is a compact $R$-orientable n manifold, then the boundary map $\partial : H_n(M,\partial M;R) \to H_{n-1} (\partial M)$ sends a fundamental class for $(M,\partial M)$ to a…
user63310
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Hatcher 3.3 Exercise 6

The exercise on Hatcher's Algebraic Topology goes: 6.$~$Given two disjoint connected $n$-manifolds $M_1$ and $M_2$, a connected $n$-manifold $M_1\# M_2$, their connected sum, can be constructed by deleting the interiors of closed $n$-balls…
Shana
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Fundamental Group of $\mathbb{RP}^n$

I was tring to culculate the fundamental group of $\mathbb{RP}^n$ with VAN KAMPEN to have a better understanding on how to use this theorem. $\left(\mathbb{RP}^n:=S^n/ \sim \left((x_0,\cdots,x_n) \sim(-x_0,\cdots,-x_n)\right)\right)$ By consider the…
kingzone
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Cellular Homology of the 3-Torus

I'm trying to compute the homology of the 3-torus $T^3=S^1 \times S^1 \times S^1$. Trying to use the typical construction the 2-torus $T^2$ as a starting point, I identified pairs of opposite faces on a cube as shown below: This gives a single…
leeabarnett
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Why is a path-connected topological space homotopy equivalent to the classifying space of its loop space?

Given a path-connected topological space $X$ (lets say compactly generated; this entire post will be working in the category of compactly generated topological spaces) with a designated point $x$, we can form the loop space $\Omega X$ of pointed…
Nehsb
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