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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.

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Are infinite Grassmanians simple spaces?

Let $Gr_n=BO(n)$ be the real infinite grassmanian. I want to know wether $\pi_1(Gr_n)=\mathbb{Z}/2$ acts nontrivially on the higher homotopy groups of $Gr_n$. It is known that the action can be computed by inspection of the action of $O(n)$ on…
Fabio Neugebauer
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Wrong proof: Hopf fibration is 2-torsion (solved)

We've been looking at a proof that the suspension of the Hopf map $\eta: S^3 \to S^2$ is an order 2 element in $\pi_4(S^3)$. In the course of that we were looking at that diagram: One can see that $\sigma$ has degree 1 while $\tau$ has degree $-1$.…
Mathis
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Homotopic maps induce isomorphic pullbacks of a principal bundle. How functorial can this be?

More precisely, I'm trying to show that the groupoid $\mathscr{B}G(X)$ of principal $G$-bundles over $X$ and isomorphisms is equivalent to $\Pi_1(BG^X)$. It seems like the right direction to try to construct my functor is $\Pi_1(BG^X) \rightarrow…
Aaron Mazel-Gee
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Degree and maps between closed orientable surfaces

Let $M_g, M_h$ be closed orientable surfaces of genus $g,h$ respectively. If $g>h$, we know there exists a map $M_g \rightarrow M_h$ of degree 1: just think of $M_g=M_h\#M_{g-h}$ and consider the map $M_g=M_h\#M_{g-h} \rightarrow M_h$ that pinches…
Luc
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Degree of maps $S^n \to S^n$ using local degree formula - practicality of determining signs

Let $X = S^n$ and $Y = S^n$ and let $f: X \to Y$ be a continuous map. Suppose that for some $y \in Y$, the preimage $f^{-1}(y)$ consists of finitely many points $x_1, \dots , x_m$. Let $U_1, \dots, U_m$ be disjoint neighbourhoods of $x_1, \dots,…
Kenny Wong
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About coverings given by orbit spaces

The question was arisen from doing Hatcher 1.3.24 (b). I would like to know what is wrong with the following. Let $X$ be a space that is path-connected and locally path-connected and $G$ be a group that makes $X \twoheadrightarrow X/G$ a normal…
Gil
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The continuous function $f: S^1 \rightarrow S^1, f(z) = z^3,$ has no sections

A section of a continuous function $f : X → Y$ is a continuous function $g : Y → X$ such that $f ◦ g$ is the identity on $Y$ . Let $S^1$ to be the unit circle in $\mathbb C$. Prove that the map $f : S^1 → S^1$ given by $f(z) = z^3$ has no…
user2345678
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Show that there are not retractions $r:X \to A$ when $X=S^1 \times D^2$ and $A$ is the circle shown in the following picture.

Show that there are not retractions $r:X \to A$ when $X=S^1 \times D^2$ and $A$ is the circle shown in the following picture I'm reading a solution for this which I don't really understand. The parts I'm not figuring out are first why is $A…
Emil Grönberg
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Does the hairy ball theorem follow from Borsuk-Ulam?

The proofs I have seen for the hairy ball theorem all use either degree of a map defined in e.g. by homology or direct computations using stereographic projections in order to use homotopy arguments in $\mathbb R^2$. Isn't there a trick to deduce…
Jochen
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Two definitions of simply connected

I'm showing that the first definition here implies the second (the other implication is obvious). My thoughts: Let $p,q$ be two paths in the space $X$. Then since $X$ is path connected there are two paths $f,g$ connecting the two endpoints of $p,q$…
snailspace
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Computing the homology groups of the torus or a cell complex

I've found this table of homology groups of the tori $T^n$. My question is: How did they compute these? More generally: what's the "recipe" to compute the homology group of say, a cell complex? Many thanks for your help!
Rudy the Reindeer
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A homotopy equivalence between spaces $B\Gamma$ and $K\Gamma$ for a graph of groups on a graph $\Gamma$

In Hatcher's "Algebraic Topology" (p. 92), the space $B\Gamma$ (for a graph of groups on a graph $\Gamma$) is defined to be a collection of spaces $BG_v$ for each vertex $v$, which are connected by certain mapping cylinders corresponding to the edge…
alephzero314
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If fundamental group of a compact path connected space is finite then its universal cover is compact

Let $X$ be a compact and path connected space and let $p:\tilde X\rightarrow X$ be its universal cover. I can show that if $\tilde X$ is compact then $\pi_{1}(X)$ is finite: $\pi_{1}(X, x_{0})$ acts on $p^{-1}(x_{0})$ via…
Ergin Süer
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Why is it important to have a classifying space for principal $G$-bundles over a based space $X$?

For a group $G$ its classifying space $BG$ can be thought of as a space with the property that $G∼ΩBG$ (the based loop space of $BG$). This actually works more generally for spaces equipped with a multiplication that is 'highly homotopically…
Jack Rock
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Torus removed two points is homotopy equivalent to wedge sum of three circles

What is an easy way to see that torus with two points removed is homotopy equivalent to wedge of three circles? I am trying to see it by viewing the torus as a square with two sides identified, but the intuition isn't clear to me, unlike the case…
Phil
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