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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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generalization of the Jordan curve theorem

Here is a plausible generalization of Jordan curve theorem which I couldn't find a rigorous proof for it. Let $K$ be a compact subset of $\mathbb{R}^2$ which is homotopic equivalent to $S^1.$ Prove that $\mathbb{R}^2-K$ has two connected components,…
Ehsan M. Kermani
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$M$ is a compact manifold with boundary $N$,then $M$ can't retract onto $N$.

There is hint: Prove $H^{n-1}(N) \to H^{n-1}(M)$ is trivial. Just don't know how to prove this.
henry
  • 309
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Geometric realization of a simplicial set is a CW-complex

I am reading the book "Simplicial objects in algebraic topology" by Peter May and I am trying to understand its proof for Theorem 14.1 on page 56 which says that the geometric realization of a simplicial set is a CW-complex. In this book the proof…
Pouya Layeghi
  • 1,470
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Fundamental group of two spheres and circle

Let two $S^2$, $X_1 = S^2, X_2=S^2$ and $X_3 = S^1$. Let $N_i,S_i$ poles of $X_i$, $i = 1,2$ and $n,s$ poles of $X_3=S^1$. Consider the space that $S^1$ is between the two spheres and glued like $N_1 \sim n \sim N_2$ and $S_1 \sim s \sim S_2$. Im…
hal97
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Mappings from $S^n$ to $S^n$ with odd degree

Let $f:S^n\rightarrow S^n$ be of odd degree, i.e. $f^*(1)$ is odd where $f^*:H_n(S^n)\rightarrow H_n(S^n)$ is the induced map on homology. Prove that there exists an $x\in S^n$ with $f(-x)=-f(x)$. I tried to imitate the proof of Borsuk-Ulam theorem,…
Anonymous Coward
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Homology of tori and the Universal Coefficient Theorem

I'm working on something that involves tori, and specifically I'm looking for the first homology group of the $n$-torus with coefficients in $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$. However, the Universal Coefficient Theorem and the algebraic tools…
Gerben
  • 796
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Is there something more to algebraic topology than detecting holes?

I have been reading in the last few months John Lee's Introduction to topological manifolds and Rotman's Introduction to algebraic topology. The reason I started learning AT is that it seemed like a beautiful and elegant theory, describing the…
yoyo
  • 109
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Singular cohomology for spaces with a group action

Consider the usual functor that associates to each topological space its singular chains \begin{align} \textbf{Top} &\to \text{Ch}(\textbf{Ab}) \\ X &\mapsto C_{\bullet}(X). \end{align} If we consider topological spaces with an action of a…
Eduardo de Lorenzo
  • 366
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Loop spaces and $A_\infty$ structure

There is a theorem saying that for any space $X$ which is connected and has the homotopy type of a CW complex, $X$ has the homotopy type of a looped space $\Omega Y$ if and only if $X$ has an $A_\infty$ structure. It is said that this result follows…
142857
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¿Is the image of a path homotopy simply connected?

I was just introduced to algebraic topology in class, we've learned about homotopies, covering maps and deformation retracts. We've also calculated some fundamental groups. I was wondering, asume you have a loop $\alpha : I \longrightarrow X$ based…
UCL
  • 361
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$X\vee Y\cong (X\times\{y_0\})\cup (\{x_0\}\times Y)$

Let $(X,x_0),(Y,y_0)$ be two pointed topological spaces. Then $X\vee Y$ obtained by $X\amalg Y/x_0\sim y_0$ where $X\amalg Y$ denote the disjoint union of topological spaces and $X\times\{y_0\}\cup \{x_0\}\times Y\subset X\times Y$: subspace are…
one potato two potato
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Understanding Hatcher's Proposition 1.26

First, let me state the proposition in Hatcher's textbook (a) If $Y$ is obtained from $X$ by attaching 2-cells as described above, then the inclusion $X\hookrightarrow Y$ induces a surjection $\pi_1(X,x_0)\to\pi_1(Y,x_0)$ whose kernel is $N$. Thus…
one potato two potato
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Equality condition from functions between the disk and the sphere

Let $f,g:D\rightarrow \mathbb{S}^2$ where $D$ is the disk $\{(x,y)|x^2+y^2\le 1\}$, such that for $(x,y) \in \mathbb{S}^1$ these functions are given by: $f(x,y)=(x,y,0)$ and $g(x,y) = (-y,x,0)$ show that there is a point $(x,y) \in D$ such that…
David Melo
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If $M^{2n}$ is oriented, then $H_n(M)\to H_n(M,\partial M)$ can be identified with the matrix of the intersection form

If $M$ is an oriented $2n$-dimensional manifold with boundary $\partial M$, then the natural map $H_n(M)\to H_n(M,\partial M)$ can be identified (via Lefschetz duality) with a map $H_n(M)\to H^n(M)$. If, moreover, $H_{n-1}(M)$ is torsion-free or we…
pancini
  • 19,216
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The product of a cofibration with an identity map is a cofibration

This is a problem from the book "modern classical homotopy theory" which I can't solve. Let $i : A \rightarrow X$ be a cofibration and $Y$ any space. Show that $i : A\times Y \rightarrow X\times Y$ is also a cofibration. I am supposed to use the…
Ben
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