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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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Torus construction

I understand the usual way to construct a torus. i.e., pasting opposite edges of a rectangle. But I don't know the construction by saying take a zero cell and attach a two one cell and then attaching a two cell. Also construction of a orientable…
Kannan
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mapping torus eqivalent definition

Let $X$ be a topological space and $f:X\to X$ a homeomorphism. I need to find a continuous, properly discontinuous $\mathbb{Z}$-action on $X\times\mathbb{R}$, such that the quotient $(X\times\mathbb{R})/\mathbb{Z}$ is homeomorphic to the mapping…
blst
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how to prove Euler Characteristic of cw complex $\chi(X)=\chi (A)+\chi (B)-\chi (A \cap B)$.

If a finite CW complex $X$ is the union of sub complexes $A$ and $B$, show that $\chi(X)=\chi (A)+\chi (B)-\chi (A \cap B)$. some how I can imagine what is happening,it is counting numbers of all kind of holes and calculate the sum of them with…
kpax
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Homology groups of 2-disc with two circles cut out and their clockwise oriented boundaries identified...

Edit: Nevermind, I believe I found all of my mistakes, so I just need to redo the computation. I am working on Hatcher 2.2.9 c) He asks us to find the homology groups of 2-disc with two circle cut out and their clockwise oriented boundaries…
Elle Najt
  • 20,740
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How to prove the zeroth homology in the long exact sequence of associated to (A,X,X/A) ...

(I am working out of Hatcher. This is theorem 2.13.) I am brewing up some confusion about the long exact sequence of the homology groups of $A \subset X$ and $X / A$. (For (X,A) a good pair, which is to say, that A is a closed subspace of X…
Elle Najt
  • 20,740
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Do $\pi_1$-surjective maps of degree $0$ exist?

A well-known theorem asserts that degree 1 maps induce surjections of the fundamental group. I am looking for a partial converse. Is it true (under suitable assumptions) that a map between compact, aspherical manifolds of the same dimension has…
user39082
  • 2,124
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Commutative Diagrams and their relationship to induced homomorphisms

Throughout my topology class my professor has used commutative diagrams on various occasions to prove results such as 1) There exists no antipode preserving, continuous, onto map, $f: S^2 \to S^1$ 2) $[0,1] \setminus 0 \sim 1$ is homeomorphic to…
user7090
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Calculating fundamental group of adjunction space with linear transformation.

$X = D^{2} \times S^{1} \cup_{f} S^{1} \times D^{2}$, where $f : S^{1} \times S^{1} \to S^{1} \times S^{1}$ is a map induced by the linear map on $\mathbf{R}^{2}$ given by the matrix $$\left( \begin{array} {cc} a & b \\ c & d \end{array} \right)$$…
fiverules
  • 777
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Clarifying the definition of regular simplicial action

I'm currently studying Chapter 3 of the book "Introduction to compact transformation groups" by Bredon. The main matter that is discussed in this chapter are simplicial actions of a finite group $G$ over a simplicial complex $K$ (i.e., group actions…
Matías
  • 263
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$H$ a discrete subgroup of topological group $G$ $\implies$ there exists an open $U\supseteq\{1\}$ s.t. the $hU$ are pairwise disjoint

Problem: Let $G$ be a topological group (i.e., $G$ is a topological space and $G \times G\rightarrow G$ is continuous), and let $H$ be a discrete subgroup of $G$. Prove that there is a neighborhood, $U$, of the identity, $1$, such that the sets $h…
user125103
  • 523
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What is the $H_1(X)$ where $X$ is the boundary of the unit square and the vertical lines with rational first coordinate?

Suppose $X$ is the four edges of the unit square, together with the vertical line segments $L_q=\{(q,y):y\in I\}$ for $q\in\mathbb{Q}\cap I$. I'm trying to compute $H_1(X)$ in particular. I think of setting $U$ to be the top three-fourths of $X$,…
Geovanna Anthony
  • 639
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Query about proof of homotopy lifting property

Let $p : Y \to X $ be a covering projection. Given a path $ u : I \to X $ and a point $ y \in Y $ with $p(y) = u(0) $, there exists a unique path $ \hat{u}:I \to Y $ with $p \hat{u} = u$ and $ \hat{u}(0) = y$. The proof I have of this proceeds as…
Martin P
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homeomorphism $T: X \rightarrow X$

How do i prove that $Tz=\bar{z}+1+i$ defines a homeomorphism $T: X \rightarrow X$ where $X=\mathbb{R}\times[0,1] \subset \mathbb{C}$ ? (how can there be a continuous bijection in this case?) Also, how do I show that if G is the group of…
user135541
  • 131
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regular values of PL maps

In the category of smooth manifolds and maps, $y$ is a regular value of $f$ iff the tangent map $df(x)$ is surjective for any $x\in f^{-1}(y)$. Then the preimage $f^{-1}(y)$ is a smooth submanifold. What is the analogical definition in the…
Peter Franek
  • 11,522
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Proving that the fundamental group of Klein bottle is generated by two elements without using covering spaces and van Kampen theorem, etc.

This is a problem in 'Topology and Geometry' by Bredon. I tried hard on this problem, but have no idea what to do. This question was onced asked by someone, but there was no satisfactory answer. Let $\mathbb{K}^2$ be Klein bottle. The problem asks…
Wooyoung Chin
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