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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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Singular homology of disjoint union of discs quotient boundaries

I am trying to compute the singular homology of the space $X=D \coprod D\setminus \sim$ with $D$ the two dimensional disc and $\sim$ the equivalence relation that identifies the boundaries of the discs. On one side I know that the homology of the…
user404720
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Prove that if $f$ is not surjective, then $f$ is homotopic to a constant via a homotopy that fixes a point

I'm trying to prove that if $f:S^1 \rightarrow S^1$ is not surjective, then $f$ is homotopic to a constant function via a homotopy that fixes a point $\theta \in S^1$. Showing that it is homotopic to a constant function is simple, but showing that…
Derek Adams
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Proving $H_i(S^n - h(D^k))=0$ $\forall i>0$

In class today we showed that $H_i(S^n - h(D^k))=0$ where $h(D^k)$ is the embedding of the k-disc into $S^n$. The proof was very technical and used induction. I am wondering why the following argument doesn't work: Since $D^k \cong h(D^k)$ and we…
user637978
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The fact that $H_n(S^n,S^n-\{x\})\cong\mathbb{Z}$ is a bit unintuitive to me, can anyone give me some insight?

The fact that $H_n(S^n,S^n-\{x\})\cong\mathbb{Z}$ is a bit unintuitive to me, can anyone give me some insight? This is a common fact in any introduction to algebraic topology course, it is achieved by using an exact sequence of pairs and then…
user637978
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Showing that a space does not deformation retract

What is a good way to show that a space does not deformation retract onto something? For example, Show that the Mobius strip does not deformation retract onto its boundary Show that the torus does not deformation retract onto the one-point union of…
user500144
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Fundamental Group of Wedge Sum: Does Munkres Assume Path Connectedness?

I'm attempting to solve exercise 71.2 of Munkres, and I have only one small roadblock remaining. Suppose $X$ is a space that is the union of the closed subspaces $X_1, \dots, X_n$; assume there is a point $p$ of $X$ such that $X_i \cap X_j = \{p\}$…
Itserpol
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Proof of that two liftings of a function are equivalent

Let $f:X\rightarrow S^1$ be a continuous function for $X$ connected and $x^* \in X$. Given that $\hat{f}_1, \hat{f}_2:X \rightarrow \mathbb{R}$ are liftings of $f$ such that $\hat{f}_1(x^*) = \hat{f}_2(x^*)$, I am trying to prove that $\hat{f}_1(x)…
user312437
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Defining a Homotopy, Continuity of a Homotopy

Problem: Let $f$ be a path from $a$ to $b$. Show that $g$ defined by $g(x)= \left\{ \begin{array}{ll} f(2x) & x \in [0, \frac{1}{2}] \\ b & x\in[\frac{1}{2}, 1] \\ \end{array} \right. $ is path-homotopic to $f$. My Solution: Define $H :…
user641658
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The functoriality of the cellular homology

Given $\pi：S^n \to \mathbb RP^n$ the usual quotient map, I am trying to figure out what is the induced map on the $nth$ homology. If I use the singular homology it seems hopeless, since it is hard to locate an explicit generator. So I decide to use…
zach
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Is there a bijection between $\pi(X,x_0)$ and the set "classes of paths from $x_0$ at $x_1$?

Let $x_0$ and $x_1$ two points of the same arc component of $X$. Is there a bijection between $\pi(X,x_0)$ and the set "classes of paths from $x_0$ at $x_1$"? How can I construct it?
Nicole
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Extending loop to compact orientable genus $g$ space $\Sigma_g$.

I am working on the following problem. $X$ is any space. Let $f:S^1\rightarrow X$ be a loop. Show that $[f]=0$ in $H_1(X)$ if and only if $f$ extends to a map $F:\Sigma \rightarrow X$ where $\Sigma$ is some compact orientable genus $g$ surface with…
Lev Bahn
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Determine the sign of local degree

In Hatcher's P136, the local degree of a map $f:S^n\to S^n$ is the degree of the map $f_*:H_n(U,U-x)\to H_n(V,V-y) $ where $U$ and $V$ are neighbourhoods of $x$ and $y$ such that $f(U)\subset V$ and $f(U\setminus\{ x\})\subset V\setminus \{ y\}$.…
anon
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Fundamental theorem of algebra using degree theory

This is problem 8-5 in Lee's Topological Manifolds: Prove that every nonconstant polynomial on one complex variable has a zero. [Hint: if $p(z)=z^n+a_{n-1z^{n-1}}+ \dots + a_0$, write $p_\epsilon(z)=\epsilon^np(z/\epsilon)$and show that there exists…
Lotte
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Proof that a certain section of the orientation cover is continuous

I am wondering whether I on the right track to show continuity of a certain function. Here is some background: Let $M$ be an $n$-manifold. Given a subset $A \subset M$ and $\alpha \in H_n(M,M\backslash A)$, define $\alpha_x$ to be the image of…
user620793
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Path Homotopy of Concatenated Paths

If I have two concatenated paths $f_{1} \cdot g_{1}$ and $f_{2} \cdot g_{2}$ where $f_{1} \cong f_{2}$ and $g_{1} \cong g_{2}$, is it fair to say $$f_{1} \cdot g_{1} \cong f_{2} \cdot g_{2}$$ by the homotopy $(f \cdot g)_{t} = (1-t)(f_{1} \cdot…
ABC
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