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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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Local contractibility of CW complex

I was trying to understand the notion of CW complex from wikipedia. The very first non-example is: $$\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subset \mathbb R^2$$ This is not a CW complex, supposedly because it is not locally…
Cerolein
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Groups Associated to Knots

Let $2
Beginner
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is it necessarily true that the map on $\pi_1$ induced by f is onto?

This is a qual question and I have no idea how to begin. Help is much appreciated. Let $X$ and $Y$ be path connected spaces, and let $f : X \rightarrow Y$ be a continuous map. The mapping cone $Cf$ of $f$ is defined to be the quotient space of $Y…
kingkongdonutguy
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Cohomology ring of a direct sum via Poincare duality

I'm trying to solve exercise 3.3.26 in Hatcher's Algebraic Topology: Compute the cup product structure in $H^{*}((S^{2}\times S^{8})\#(S^{4}\times S^{6});\mathbb{Z})$, and in particular show that the only nontrivial cup products are those dictated…
PeterM
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Classification of covering spaces

I just found this document http://www.math.toronto.edu/~drorbn/classes/0405/Topology/CoveringSpaces/CoveringSpaces.pdf in which it is said that we can use the classification of covering spaces theorem : "Theorem: If $B$ is connected and locally…
user15542
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every mapping from $\mathbb{S}^2$ to $\mathbb{R}$ has three mutually perpendicular vector with same value

Prove that every continuous mapping $f$ from $\mathbb S^2$ to $\mathbb R$ has three vector which are mutually perpendicular and have same value i.e $f(v_1)=f(v_2)=f(v_3)$
Mahan
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Covering spaces of $\mathbb{RP}^n\times\mathbb{RP}^n$ for $n > 1$.

The fundamental group of $X = \mathbb{RP}^n\times\mathbb{RP}^n$ is just $G=\mathbb{Z}_2\times \mathbb{Z}_2$ when $n > 1$. So connected coverings of $X$ correspond to subgroups of $G$. This has $5$ subgroups: the trivial subgroup, $G$ and $3$…
curious
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Loop is contractible iff it extends to a map of disk

Let $X$ be a space, $f : S^1 \to X$ be continuous function. $f$ is homotopic to constant map $h = c$ iff $\exists$ continuous $g: D^2 \to X $ such that $g |_{S^1} = f $ My Attempt Take a homotopy $F: S^1 \times I \to X $ from constant map $c$ tot…
ILoveMath
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Lifting correspondence in Algebraic topology

Recently I have been studying algebraic topology and came across the notion of lifting correspondence. Here, lifting correspondence definition is the same that Munkres uses in his book. However, I cannot get any intuition behind that notion. I am…
James Bond
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$R$-orientable manifolds and some covering spaces associated to them

I think I have solved it, but not sure however. The "solution" is in the most recent update. Currently I'm reading through Hatcher's Algebraic Topology, and I'm stuck on one thing in the text. At page 236, he states that if $M$ is a closed connected…
Dedalus
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Is Map($T^4$,$S^2$) connected?

Consider the set $Map(T^4,S^2)$ of continuous maps from the 4 dimensonal torus $T^4$ to the 2 dimensional sphere $S^2$, endowed with compact-open topology, can we show it is not connected? How can we calculate its singular homology and $\pi_1$?
user93417
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Pontrjagin square (Mosher and Tangora Question)

This is an elaboration on a question/answer posted on MathOverflow. As per the MO question, the actual question is (with the typo corrected) Suppose the cocycle $u\in C^{2p}(X;Z)$ satisfies $\delta u=2a$ for some $a$. i. Show that $u \cup_0 u +…
Juan S
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Fundamental Polygon of Real Projective Plane

Wikipedia gives the following fundamental polygon for the real projective plane $\mathbb{R}\mathrm{P}^2$ The problem here is that the corners aren't identified to a single point (like in the fundamental polygon of the torus). I don't think this…
PeterM
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Fill a blank - Algebraic Topology

I am trying to solve the following problem in Allen Hatcher's Algebraic Topology book : Let $A_1, A_2, A_3$ be compact sets in $\mathbb R^3$. Use the Borsuk-Ulam theorem to show that there is one plane $P \subset \mathbb R^3$ that simultaneously…
Patrick Da Silva
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connected sums of closed orientable manifold is orientable

a general version: connected sums of closed manifold is orientable iff both are orientable. I think this can be prove by using homology theory, but I don't know how.Thanks.
henry
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