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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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Homology of product of topological space and sphere is direct sum of homologies.

Show that for $i > n \in\mathbb{N}$: $$H_{i}\left(X \times \mathbb{S}^{n}\right) \simeq H_{i}\left(X\right) \oplus H_{i - n}\left(X\right).$$ My first idea motivated by $n=0$ case (which is obvious) was to try induction but I cannot see how to…
Stephen Dedalus
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Problem 1.1.5 from Hatcher's

I want your advice on my solution of this problem. problem Show that for a space $X$, the following three are equivalent: (a) Every map $S^1 \rightarrow X$ is homotopic to a constant map, with image a point. (b) Every map $S^1 \rightarrow X$ extends…
MathematicalPhysicist
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Why $T\mathbb{S}^{2}\otimes T\mathbb{S}^{2}$ is trivial?

I encountered this problem in an homework problem set of algebraic topology. Naturally I thought about Bott Periodicity which implies $K(\mathbb{S}^{2})\cong \mathbb{Z}$. But I am beware that we are working in $KO$ theory instead of $K$ theory, so…
Kerry
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Hopf Invariant Definitions

I have seen two definitions of the Hopf invariant given: (1) Cohomological Definition: Let $S^{n}$ denote the oriented $n$-sphere, where $n \geq 2$. Let there be given a map $f:S^{2n-1} \rightarrow S^{n}$. Consider $S^{2n-1}$ as the boundary of an…
user 3462
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Homotopy between two functions to a circle.

Suppose $f,g: X\to S^1$ are such that $f(x)\neq -g(x)$ for any $x\in X$. I need to construct a homotopy between these two functions. Now, the fact that $f(x)\neq -g(x)$ guarantees that there is always a unique shortest path between $f(x)$ and $g(x)$…
Jimmy R
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the fundamental group of a Riemann surface with n points removed

Can anybody give me some suggestions about how to calculate the fundamental group of a Riemann surface with $n$ points removed? For example, what is the fundamental group of the genus $1$ torus with $4$ points removed? Thanks.
Qiao
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Zero Cohomology means zero homology?

Suppose we have a space $X$, which has zero cohomology (except in degree zero). Does he neccesarly have zero homology (except in degree zero)? If not, what if $X$ is a manifold? Universal Coefficient theorem gives me, that $Hom(H_*(X),\mathbb{Z})$…
Tina
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Homology of Euclidean space $H_n(\mathbb{R}^m)$

I've been computing some singular homology groups of different spaces. In particular, I know how to compute the homology of a cell complex. Now I'm wondering how to compute the homology of $\mathbb{R}^m$. Since homology is a way of counting holes…
Rudy the Reindeer
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weak homotopy equivalence (Whitehead theorem) and the *pseudocircle*

On wikipedia, I recently read about a highly pathological finite topological space, namely the pseudocircle $$X=\{a,b,c,d\},\;\;\; \mathcal{T}=\{\emptyset,\{a\},\{b\},\{ab\},\{a,b,c\},\{a,b,d\},X\}.$$ It is stated that the map $$\begin{array}{c r c…
Leo
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How to read homotopy schematics?

I am attempting to start working through J.P. Mays A Concise Course in Algebraic Topology but can't seem to understand what it describes as "schematic indications" of how a given homotopy behaves on "the domain squares." He first defines three…
Joseph Weissman
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Examples of $X$ such that $\pi_1(X) $ not abelian?

I've come up with some examples to apply the Hurewicz theorem to compute $H_1(X)$. This is only interesting if $\pi_1(X)$ is not abelian. The only examples of $X$ such that $\pi_1(X)$ not abelian I can come up with are $\vee_i S^1$ and $\Sigma_g$…
Rudy the Reindeer
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Spaces such that $\Omega^2 X \cong X$

We know from Bott Periodicityt that there is a space X such that $\Omega^2 X \cong X$ (homotopy equivalence) , but these spaces are rather complicated and I am curious, is there any easy example of a non-contractible, path-connected space X such…
Heidar
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Using Mapping cone to show map induce isomorphism on homology

In Hatcher's Algebraic Topology Corollary 3A.7(about p266), he seemed to used a fact that if a map whose reduced homology of the mapping cone are all zero , then it induces isomorphism on the homology. Can anyone help me to understand this?
user93417
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What type of topological space is this?

I came a cross an exercise the other day considering the following quotient space: Let $T$ be a torus and let $A, B \hookrightarrow T$ be two parallel circles. Let $X$ be the quotient space collapsing all of $A$ to a point, and all of $B$ to a…
mNugget
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Retractions and fundamental groups

On page 39 of Allen Hatcher's Algebraic topology (page 19 in the pdf document available on Allen Hatcher's website), exercise 16 c), I have the following proof : Let $\varphi : S^1 \to S^1 \times D^2$ be a parametrization of the displayed curve.…
Patrick Da Silva
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