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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Cell Complex of Cartesian Product

Let $X$ and $Y$ be cell complexes. Then $X \times Y$ has the structure of a cell complex with cells the product of $e^{m}_{\alpha} \times e^{n}_{\beta}$ where $e^{m}_{\alpha}$ ranges over the cells of $X$ and $e^{n}_{\beta}$ ranges over the cells of…
user404735
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Projective plane and homotopies

If we consider $P^2$ (the real projective plane) like a disk with its boundary identified via the antipodal map, we can consider, in this model, the 180 degrees rotation. It's easy to see that this rotation and the identity are free-homotopic. Are…
hal97
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Confusion about p-completion in Adams spectral sequence

In section 3.1 of Complex Cobordism and Stable Homotopy Groups of Spheres, Ravenel computes the homotopy groups of $MU$ using the Adams spectral sequence. He comes to the conclusion that the $E_2$ page is $C \otimes P(a_0, a_1,\dots)$ where…
Connor Malin
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Basic computation of a double graded spectral sequence: $^I E^0_{pq}$

Let $C=\oplus_{p\geq0, q\geq0}C_{pq}$ be a double graded group with two differentials: $d^I_{pq}:C_{pq}\rightarrow C_{p-1,q}$ and $d^{II}_{pq}:C_{pq}\rightarrow C_{p,q-1}$, with the usual assumption on the anticommutator of $d^I, d^{II}$ being 0,…
gmoss
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Homology relative to a point,

Let X be a topological space, and $P \in X$ a point. I had to prove $\tilde H_{i}(X) \simeq H_{i}(X,P)$. Instead of working on the complex i tried to use some fact of group theory but I'm not so sure of the validity of my conclusion. Is this…
Augusto Matteini
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1-cell on path connected space is homotopy equivalent to wedge

Suppose $X$ is a path-connected space, and we attach a 1-cell to it with some attaching map $f : \{0,1\} \to X$ and call the resulting space $Y$. Is $Y$ homotopy equivalent to $X \vee \mathbb{S}^1$?. My idea was the following: let $g : [0,1] \to X$…
kabosu
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Attaching a cell

Could you help me to explain this argument: Let $f: S^{n-1} \rightarrow A$ for $n \ge 1$, form $$X= C(f) := \dfrac{A\coprod D^n}{f(x) \sim x, \forall x \in S^{n-1}}$$ "$(D^n,S^{n-1}) \rightarrow (X,A)$ induces isomorphisms in $H_q(D^n,S^{n-1})…
user69833
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What map of ring spectra corresponds to a product in cohomology, especially the $\cup$-product.

Let $E$ be a spectrum, $X$ a CW-complex and associate a graded abelian group $$ E^*(X)=\bigoplus_{k\in\mathbb{Z}}[X_+,S^k\wedge E] $$ to it. The brackets denote stable homotopy classes. Please let me ignore the suspension spectrum symbol…
Ronald Bernard
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Forgetting basepoints map is neither injective nor surjective.

Let $X$ and $Y$ be topological spaces with basepoints $x_0 \in X$ and $y_0 \in Y$. Forgetting basepoints defines a map $$\Phi:[(X,x_0),(Y,y_0)]_∗ →[X,Y].$$ We write $[X, Y ]$ for the set of homotopy classes of continuous maps from $X$ to $Y$. And…
Gillyweeds
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The boundary of cells in a CW complex

One of axioms of the definition of a CW complex says that the boundary of a cell in contained in a finite union of cells of low dimension. Here, all cells are open cells. My question is: Given a CW complex, is the boundary of its cells is exactly a…
G.-S. Zhou
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Finding all connected covering spaces of the circle $S^1$

I want to solve this problem that I found in a qualifying topology exam: "Let $S^1$ be the unit circle in the complex plane. How many isomporphism classes of connected covering spaces of $S^1$ exist? Construct a representant of each class." The…
Twnk
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Van Kampen counterexamples

The van Kampen theorem states that if I have a space $X = \cup_\alpha A_\alpha$ where $A_\alpha$ are all path connected and if for any given $\alpha, \beta, \gamma $ we get $A_\alpha \cap A_\beta \cap A_\gamma$ is also path connected then the…
YankyL
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Is $a \mapsto [a \cdot id]: A \to H_n(\Delta^n, \partial \Delta^n;A)$ an isomorphism?

I am hoping for a problem I am working on that the function defined above is an isomorphism of groups, but I cannot manage to prove it. Is $[a \cdot \sigma] = [a \cdot id: \Delta^n \to \Delta^n]$ for any $\sigma: \Delta^n \to \Delta^n$?
blacksmith
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$H_k(E,\mathbb{Z})\cong H_k(X,\mathbb{Z})$

Let $\pi:E\rightarrow X$ be a real vector fibrate of rank $n$. How to prove that $$H_k(E,\mathbb{Z})\cong H_k(X,\mathbb{Z}).$$ approach: I consider it true that $X$ is a deformation rectract of $E$, it is true? Any hint would be appreciated.
felipeuni
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A map from $S^{1}\times\cdots\times S^{1}$ to $S^{n}$ with nonzero degree

Can you construct a map $$F: S^{1}\times\cdots\times S^{1}(n~\text{copies of}~S^{1})\rightarrow S^{n}$$ of nonzero degree?
shrinklemma
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