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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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Why is this entangled circle not a retract of the solid torus?

I'm doing exercise 16 on page 39 in Hatcher: Show that there are no retractions $r: X \rightarrow A$ in the following cases: (a) $X = \mathbb{R}^3$ with $A$ any subspace homeomorphic to $S^1$ (b) $X = S^1 \times D^2$ with $A$ its boundary torus…
Rudy the Reindeer
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Degree in homology and cohomology

Let $f : S^n \rightarrow S^n$ be a continuous map. One can define the degree of $f$ as the integer $k$ such that $f_* : H_n(S^n) \rightarrow H_n(S^n)$ is multiplication by $k$ on $H_n(S^n) \cong \mathbb Z$. However, it also makes sense so define the…
user15464
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How should I interpret a homotopy schematic?

I'm having trouble really making sense of these homotopy schematics from May's A Concise Course in Algebraic Topology Here's what I understand so far: - The top and bottom rows are different compositions of paths, and the schematics are aiming to…
MBP
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Fundamental group of the complement of Borromean rings

I got a homework asking me to show the fundamental group of the complement of the Borromean rings (say, $S^3\setminus B$ where $B$ is the Borromean rings) I know there should be three generators, but I had a hard time to find the relations; I know…
John0417
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locally simply-connected vs. semilocally simply-connected

Can someone help me understand the difference between locally simply-connected and semilocally simply-connected? I actually need more help understanding (with an example), what it means to be locally simply-connected. I know that a space X is…
cbro
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About two space with infinite circles.

This is problem 1.2.20 from Hatcher's algebraic topology. I am able to complete the first part. But I have trouble proving they are homotopy equivalent but not homeomorphic. Can someone please give some hints? Thanks.
JSCB
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Generator for homology of surface of genus $g$ - Hatcher 2.2.29

Consider the surface $M_g$ of genus $g$, embedded in $\Bbb{R}^3$ in the standard way. It bounds some compact region $R$. Two copies of $R$ are glued together by the identity map between their boundary surfaces, which forms a closed 3-manifold $X$. I…
user38268
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Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$?

Can contractible subspace be ignored/collapsed when computing $\pi_n$ or $H_n$? Motivation: I took this for granted for a long time, as I thought collapsing the contractible subspace does not change the homotopy type. Now it seems that this is only…
user325
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Pointed cofibrations between well-pointed spaces

Recall that we call a map $i: A \rightarrow X$ a cofibration if it has the homotopy extension property. We will say a pointed space $X$ is well-pointed, if the inclusion of the basepoint $\{ * \} \hookrightarrow X$ is a cofibration. A pointed…
Piotr Pstrągowski
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Geometrical interpretation of cup product.

I know geometrically the meaning of cohomology groups of topological spaces. Is their any geometrical interpretation of cup product?
King Khan
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The first Stiefel-Whitney class is zero if and only if the bundle is orientable

Ralph Cohen's notes on the topology of fiber bundles pp.84 (theorem 3.3) says that it follows immediately from the definition of the first Stiefel-Whitney class of real vector bundles (pp.83) \begin{equation} (BSO(n) \to BO(n)) \in…
PhysicsMath
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Hatcher Algebraic Topology 0.24

This is my second question from Hatcher chapter 0 (and final I think). For $X$, $Y$ CW complexes, it asks one to show that $$X \ast Y = S(X \wedge Y)$$ by showing $$X \ast Y/(X \ast y_0 \cup x_0 \ast Y) = S(X \wedge Y) / S(x_0 \wedge y_0),$$ where…
Sina
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Poincaré Duality Example

It is intuitively clear that the Poincaré dual of a ray $\{(r,0);r>0\}$ in $\mathbb{R}^2-\{0\}$ is the form $\frac{1}{2\pi} d\theta$. For some reason I do not understand, I failed to prove this rigorously. Can somebody help me? (this example comes…
brunoh
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Equivariance of lifts to universal cover

Let's say you have a homotopy equivalence $f: X \rightarrow Y$. Consider the lift $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ to their universal covers. Let $\gamma$ be an element of the fundamental group of $X$, which can be identified…
h_a
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Map Surjective on a Disk

I've got another question from a student that has stumped me: Let $D^{n+1}$ be the $n+1$-disk, with boundary sphere $S^n$. Suppose $f:D^{n+1}\longrightarrow \mathbb{R}^{n+1}$ is a map such that $f(S^n)\subseteq S^n$. Furthermore, suppose that…
J126
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