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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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Singular homology of spaces with trivial topology

Suppose $X$ is endowed with the trivial topology, e.g. $X$ and $\emptyset$ are the only open sets. For the sake of simplicity, I'll assume that $X$ is finite, $|X| = m$. Now, the $n$-th module of the singular chain complex of $X$ should be a free…
athanasios
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Euler characteristic in pullbacks

Given a pushout $P=B\cup_AC$ which we represent as a commutative diagram $$ \begin{matrix} A & \stackrel{f}{\rightarrow} & B\\ \downarrow{g} & &\downarrow{k} \\ C &\stackrel{h}{\rightarrow} & P \end{matrix} $$ the euler characteristic is…
palio
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Higher Dimensional equivalent of genus

For 2-manifold there exists the notion of genus. I (as a non topologist) was wondering if there exists something similar for d-manifolds. Thank you
stefan
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Homotopy of singular $n$-simplices

I was wondering if there's any way to fit in homotopy into the definition of singular homology. Assume that two singular $n$-simplices are homotopic as maps, does this relate them in any way as elements of the homology group $H_n$? I guess we should…
pki
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Effect of the degree of a map $S^n\to M$ on lower homology groups

I'm looking through some old algebraic topology problems to study for an exam, and I came across the following: Let $M$ be a compact, orientable $n$-manifold, and let $f:S^n\to M$ be a map of degree $d(f)$. For $0
user161142
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Homotopy groups of $n$-torus with a point removed.

Is there a simple way how to compute and present homotopy groups of $T^n=S^1\times \ldots\times S^1$ with a point (or several points) removed?
Peter Franek
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Is $\Omega \tilde X \simeq \Omega_0 X$?

Let $\tilde X\to X$ be a universal covering of a based space $X$, with a chosen basepoint. Is $\Omega \tilde X \simeq \Omega_0 X$? Here $\Omega$ denotes the loop space, $\Omega_0$ denotes the connected component of the trivial loop and $\simeq$…
Bruno Stonek
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CW-complex of a genus n surface

I've faced some difficulty in how to equip an orientable compact connected surface of genus n with a CW-structure using the 4g-gon model. I understand how a torus is constructed: -Start with 1 point (a 0-cell) -Add two lines which their start and…
Julian Sutherland
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Degrees of maps in algebraic topology

Please can I have some tips on how to construct maps between topological spaces of a given degree? For example, how would you go about building a map of degree $3$ from $\mathbb{CP}^1\times\mathbb{CP}^2 \to \mathbb{CP}^3$? Or a map from $S^2\times…
Jeremy Judge
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Why doesn't $S^n$ embed into $R^n$?

It seems obviously true, but how does one actually show this? Or what tools does one use? I only know the basics of homotopy theory and homology. Can I use invariance of domain somehow? If $S^n$ embeds, then so does a neighborhood of it in…
Elle Najt
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Relation between homotopy groups of $S^n$ and homotopy groups of $SO(n)$, $O(n)$

What is the relation between the homotopy groups of spheres $S^n$ and the homotopy groups of the special orthogonal groups $SO(n)$ (resp. $O(n)$)? This question occurred to me in the context of classifying real vector bundles over spheres via…
Dave
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Showing $\pi_1(M) = F$ ($F$ a finitely generated free group and $M$ an $n$-manifold of dimension greater than $2$)

Problem: Let $F$ be a finitely generated free group. Prove that there is an $n$ manifold, $M$, $n > 2$ with $\pi(M) = F$. Let $F = F_S$ s.t. $|S| \in \mathbb{N}$. If I could show that there exists $n$-manifolds $M_1, \ldots , M_k$…
user1770201
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Fundamental group of the complement of a linear subspace

Let $m
user18233
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A characterization of wedges of 1-spheres and 2-spheres?

It is well known that if $X$ is a $1$-connected (i.e. path connected and simply connected) 2-dimensional finite simplicial complex, then $X$ is homotopy equivalent to a wedge of $2$-spheres. Consider the more general setting where $X$ is path…
Matthew Kahle
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Applications of Brown's Representability Theorem

I am currently trying to understand the proof of Brown's Representability Theorem, which says that any generalized cohomology theory is represented by an $\Omega$-spectrum. Can anyone point me to some interesting applications of this theorem, within…
user137689
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