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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
16
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4 answers

Antipodal map on $ S^n $ homotopic to identity map if $n$ is odd

I want to prove that the antipodal map from $S^n$ to $S^n$ is homotopic to the identity map if $n$ is odd. (I know it's actually true if and only if) If I consider the map $ H(x,t) = (1-2t)x $, why doesn't this work? I don't really see how this is…
TRY
  • 397
16
votes
3 answers

De Rham cohomology of the plane with $m$ holes

I'm trying to compute the De Rham cohomology of the manifold $$M_d=\mathbb{R}^2\setminus\{ p_1\ldots p_d\},$$ where the points $p_1\ldots p_d$ are all distinguished. This must be a standard exercise in the use of the Mayer-Vietoris sequence, but I…
Giuseppe Negro
  • 32,319
16
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2 answers

The fundamental group of a pair of Hawaiian earrings

Let $H$ be the Hawaiian earring and let $H'$ be the reflection of the Hawaiian earring across the $y$-axis (in the Wikipedia picture). There is a canonical homomorphism from the free product $\pi_1(H) * \pi_1(H')$ to $\pi_1(H \cup H')$ (with…
Qiaochu Yuan
  • 419,620
16
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3 answers

Fundamental group of $S^2$ with north and south pole identified

Consider the quotient space obtained by identifying the north and south pole of $S^2$. I think the fundamental group should be infinite cyclic, but I do not know how to prove this. If it is infinite cyclic, would this and $S^1$ be an example of two…
user6587
16
votes
1 answer

Euler characteristic of covering space of CW complex

I am trying to prove the following statement: if $X$ is a finite CW complex and if $Y \to X$ is a $n$-sheeted covering then $Y$ is a finite CW complex and $\chi(Y)=n \cdot \chi(X)$. I know the covering space of CW complex is again a CW complex, but…
user198206
16
votes
2 answers

Explicit isomorphism $\tilde{H_n}(X)\simeq \tilde{H}_{n+1}(SX)$.

Let $X$ be a topological space and $S:\mathbf{Top}\to \mathbf{Top}$ be the suspension functor. It's not hard to show using e.g. the long exact sequence of homology that $\tilde{H_n}(X)\simeq \tilde{H}_{n+1}(SX)$ (where $\tilde{H_n}$ denotes reduced…
Bruno Stonek
  • 12,527
16
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1 answer

Fundamental group of quotient spaces of $SO(3)$

I am trying to figure out the fundamental group (actually simply connected or not will suffice) of the following quotient space of $SO(3)$: Let $X = SO(3)/E$, where $E$ is the equivalence relation defined as follows: $E \equiv M \sim {S_{A}}^{i} * M…
Srikanth
  • 689
15
votes
5 answers

Reduced homology

I'm trying to understand why $H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$. In Hatcher on page 110 he writes ''...Since $\varepsilon \partial_1 = 0$, $\varepsilon$ vanishes on $im \partial_1$ and hence induces a map $H_0(X) \rightarrow \mathbb{Z}$…
Rudy the Reindeer
  • 46,359
15
votes
2 answers

Homeomorphism of the Disk

I'm working through Massey's "Basic Course in AT." One of the problems is prove that a homeomorphism of the closed disk maps the boundary to the boundary and the interior to the interior. How would one prove this? I can't seem to get this one…
SingularDegenerate
  • 521
15
votes
1 answer

Borsuk-Ulam Theorem for torus.

Is the Borsuk-Ulam theorem valid for a torus? In other words, for any map $f: S^1 \times S^1 \rightarrow \mathbb{R^2}$ there is a point $(x,y) \in S^1 \times S^1$ which $f(x,y)=f(-x,-y)$ I'm very stuck on this task. Can someone give a hint? Or…
mathmaniac
  • 534
15
votes
2 answers

Why does an orientable surface of genus >1 always have an irregular 3-fold cover?

This is a question from an old qualifying exam in topology. Let $S_g$ be the compact orientable surface of genus $g$. Show that $S_g$ has an irregular 3-fold cover when $g>1$. While the question does not explicitly state it, I am pretty sure we…
Micky
  • 316
15
votes
1 answer

Cup product and hypercohomology

I always found the cup product slightly mysterious. Recently I discovered the following interesting theorem (in Voisin's book Hodge theory and complex algebraic geometry I, chapter 4.3): For the setup, let $(X, \mathcal{O})$ be a ringed space,…
Tom Bachmann
  • 221
15
votes
1 answer

Show that $\Sigma(\mathbb{R} P^3)$ is not homotopy equivalent to $\Sigma(\mathbb{R} P^2 \vee S^3)$

I'm currently studying for my qualifying exam in algebraic topology, and I'm looking over old exam questions. The first part of the question was to show that $\mathbb{R} P^3$ is not homotopy equivalent to $\mathbb{R} P^2 \vee S^3$, which is fairly…
Robert Usher
  • 153
14
votes
3 answers

Why every smooth orientable 4-dimensional manifold admits an immersion into $\mathbb{R}^{6}$?

Why every smooth orientable 4-dimensional manifold admits an immersion into $\mathbb{R}^{6}$? This is a one-line question as I see the statement in a comment by Michael Hopkins(update: this is wrong, it is by Peter Kronheimer) . I thought about it…
Kerry
  • 2,286
14
votes
1 answer

When is $\pi_0$ a group?

In general, $\pi_0(X)$ is the set of path components of $X$ and does not have a group structure. After all, $S_0$ is just two points and the usual way of multiplying using the equation of a sphere doesn't work. But sometimes it is. For example, if…
Josh
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