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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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Inductively Constructing Chain Homotopies

Let $f,g:C_\bullet \rightarrow D_\bullet$ be chain maps. Let $T$ be a generator (possibly not the only one) of $C_n$ and assume $H_n(C)=0$. I'm trying to prove that for every $T$ there's a solution $z$ to the equation $$\partial…
user153312
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degree of a map $f:S^1\rightarrow S^1$

Let $f:S^1\rightarrow S^1$ is a continuous map with the property that $f(-x)=-f(x)$ for all $x\in X$. Question is to show that degree of $f$ is odd. We have $(\exp \circ g)(t)=f(e^{2\pi it})$ for all $t\in \mathbb{R}$. Now, $f(-x)=(\exp \circ…
user87543
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Hatcher and "house with two rooms"

On page 4 of Hatcher's "Algebraic Topology" he constructs the "house with two rooms" space. He claims that there is some neighborhood containing this space that is homeomorphic to the unit ball (in $\mathbb{R}^3$). I was hoping someone would…
wellfedgremlin
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Are open subsets of $R^n$ homeomorphic if all homotopy groups are trivial

I am wondering if the following is true: Let $U, V \subset \mathbb{R}^n$ be open sets. Suppose $\pi_i(U) =\pi_i(V) = {1}$ for all $i = 0,1,2,3, ...$ Then $U$ and $V$ are homeomorphic. I came about this question after reading Hatcher's proof…
wellfedgremlin
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Retract and its fundamental group.

Suppose that $X$ is a topological space, and $A$ is a retract with retraction $r: X\rightarrow A$ and $i:A\rightarrow X$ the inclusion map. Prove that if $i_*\pi(A,a)$ is normal then $$\pi(X,a)=\text{Im } i_* \times \text{Ker } r_* $$ I would…
EQJ
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Finding a simplicial complex with a special homological feature

I have seen the following result in a few algebraic topology texts (such as Spanier), but only as an exercise: For any sequence $m_1, \ldots, m_n$ of nonnegative integers, there is a connected simplicial complex $K$ with $H_p(K)$ free abelian of ran…
Vulcan
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Universal Cover of $\mathbb{R}P^{2}$ minus a point

I've already calculated that the fundamental group of $\mathbb{R}P^{2}$ minus a point as $\mathbb{Z}$ since we can think of real projected space as an oriented unit square, and puncturing it we can show it is homotopy equivalent to the boundary…
TinaBelcher
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Metrizability of the symmetric product of a metric space

The (infinite) symmetric product of a based topological space $(X,e)$, denoted by $SP(X,e)$, can be viewed as the topological space of ''multisets'' in $X$ containing the base point $e$ infinitely many times (please see…
Yohei Tanaka
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Understanding cellular homology: degree of attaching map of a two cell

I am working towards an understanding of cellular homology as explained here on Wikipedia. To help me I am calculating a simple example: I have two problems: good mathematical notation and actual correctness of what I'm doing. First, let me show…
a student
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Show that $M^n-N^{n-1}$ has exactly two components with $N^{n-1}$ as the topological boundary of each.

This is the problem 6.8.1. from "Topology and Geometry" by Glen E. Bredon. The problem is, If $M^n$ is a connected, orientable, and compact $n$-manifold with $H_1(M^n;\mathbb{Z}) = 0$ and if $N^{n-1} \subset M^n$ is a compact connected…
ljh8372
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Homology of universal cover of $S^1 \vee S^1 \vee S^2$ is not the same as homology of $\Bbb R^2$

I want to show, that although the homology groups of $X :=S^1 \vee S^1 \vee S^2$ and the torus $T^2$ are isomorphic, the homology groups of their universal covers are not. Let $U_X$ be the universal cover of $X$ The first part was easy since…
JBantje
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How do I informally prove this using Van Kampen theorem?

Let $X$ be the space obtained from two tori $S^1\times S^1$ by identifying a circle $S^1\times\{x_0\}$ in one torus with the corresponding circle $S^1\times\{x_0\}$ in the other. Calculate $\pi_1(X)$. Well, my professor only explains what…
Rubertos
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Union of simply connected spaces at a point not simply connected

I came across this example Spanier's Algebraic Topology book (by way of the Munkres Topology book). I kind of have an intuitive idea of why the space isn't simply connected but can't figure out a formal way to prove it isn't. The construction of…
user171177
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Proving that the orientation bundle of a non orientable manifold is isomorphic to every other oriented 2-coverings of such manifold

I've got some problem proving this statement, recalling that for me, an orientable manifold is a manifold which admits an atlas such that the transition functions have always local degree $1$ (we are working with topological manifolds). I cannot…
Luigi M
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Difference between simplex and simplicial complex

First I know the definition of simplex intuitively as follows, Simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. And the defintion of simplicial complex as follows from wiki, Simplicial complex is a…
phy_math
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