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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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If winding number of two curves are same then are they homotopic?

If winding number of two curves are same then are they homotopic ? I need this to prove $\pi(S^1)$ is isomorphic to $\mathbb{Z}$ (I know converse is true)
dragoboy
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Representation of the fundamental class of a closed orientable $n-$manifold

Let $M$ be a closed orientable $n$-manifold with a ∆-complex structure. Let ${σ_1 . . . , σ_k}$ be the set of all $n$-simplices. How does one prove that the fundamental class $[M]$ can be represented by (the cycle) $\sum\limits_{i=1}^{k}\sigma_i$ in…
adrija
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Exercise 3.4 in Rotman's An Introduction to Algebraic Topology

I am self-learning algebraic topology by reading Rotman's An Introduction to Algebraic Topology. I am stuck on Exercise 3.4 on page 41. I'd be grateful for any hints or solution. Exercise 3.4: Let $\sigma:\Delta^2 \to X$ be continuous, where…
Dan
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Universal Cover of the Punctured Torus

I am currently trying to compute the universal cover of the punctured torus. I originally thought it would be a lattice, but while computing the fundamental group I saw that the punctured torus is actually homotopy equivalent to the wedge product…
TinaBelcher
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The Oriented Universal Bundle in Characteristic classes by J.W. Milnor.

I have a problem to understand the section "The Oriented Universal Bundle" in the page 145 of "Characteristic classes" written by J.W. Milnor. The content in that page is like below, $G_n(\mathbb{R}^{n+k})$ is an unoriented Grassmann manifold. Let…
ljh8372
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Practice Problem Fundamental Group of 7-figured polygon

The question is from Munkres: Consider the space $X$ obtained from a seven-sided polygonal region by means of the labelling scheme $abaaab^{-1}a^{-1}$. Show that the fundamental group of $X$ is the free product of two cyclic groups. The solution…
ProbsNot
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What to do when this theorem can't be applied: How to calculate $H_1$?

Consider the following theorem (Lee's book on topological manifolds, page 369): (Homology Effect of Attaching a Cell) Let $X$ be any topological space and let $Y$ be obtained from $X$ by attaching a closed cell $D$ of dimension $n \ge 2$ along the…
a student
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Is it possible for the covering space of a topological space to have a fundamental group isomorphic to the fundamental group of the base?

Let $M$ and $N$ be two connected manifolds, such that $M$ is a covering space of $N$. Is it possible for $\pi_1(M)$ to be isomorphic to $\pi_1(N)$ in some non-canoncial way? EDIT: Of course, I'm excluding the trivial case where $M=N$ and the…
Braindead
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Connecting Homomorphism in LES of fibration

Let $p:E\rightarrow B$ be a Serre fibration of path connected spaces with fiber $F$. Are the connecting homomorphisms $\partial:\pi_{n+1}(B)\rightarrow \pi_{n}(F)$ in the long exact sequence of $p$ induced by a continuous map $\Omega B\rightarrow…
J.K.T.
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Homology groups of $\mathbb{R}^3 - \{C_1,C_2\}$ where $C_i$ are disjoint circles

I am reviewing for my topology final and came up with this example. I want to compute the homology groups of $X = \mathbb{R}^3 - \{C_1,C_2\}$ where $C_1$ and $C_2$ are disjoint copies of $S^1$, so basically the complement of two disjoint unknots. I…
Hubble
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How to picture a projective variety?

The picture of $\{(x:y:z) \in \mathbb P_{\mathbb C}^2 | yz =0\}$ is two spheres (each representing a copy of $\mathbb P_{\mathbb C}^1$) intersecting at one point (representing $(1,0,0)$). But why is the picture of $\{(x:y:z) \in \mathbb P_{\mathbb…
sunkist
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How can I exhibit a generator of $H^3(S^3,\mathbb Z_3)$?

How can I exhibit a generator for the third cohomology group $H^3(S^3,\mathbb Z_3)$ of the $3$-sphere with coefficients in $\mathbb Z_3$?
Mariano Suárez-Álvarez
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Lifting a sphere-valued homotopy.

Let $A\subseteq X$ be two finite cell complexes, $\dim X\leq 2n-3$ and let $[(X,A), (S^n, *)]$ be the relative cohomotopy group. There is a natural map $$ \delta: [(X,A),(B^n,S^{n-1})] \to [(X,A), (S^n, *)] $$ induced by $\pi: (B^n, S^{n-1})\to…
Peter Franek
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Another question in Bott-Tu.

I have a question regarding something on the Bott-Tu's book "Differential Forms in Algebraic Topology". At page 109, near the end, there is the following example: I don't manage to understand why the restriction $\rho_V^U$ is the…
Biagio
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Why must a map from $X$ to $S^{n}$ not be onto for it to be null homotopic

Gamelin "Topology" has an exercise 3.3.1 to prove that if $n \geq 2$ then $S^{n}$ is simply connected. Then as a hint he suggests: show that every loop in $S^{n}$ is homotopic to a loop that does not cover all of $S^{n}$. Similarly, Armstrong…
user12802
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