Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
2 answers

Prove that all contractible spaces are simply connected.

Prove that all contractible spaces are simply connected. It's simple to prove that the space is pathwise connected. But, how can I prove that the fundamental group is trivial?
Henfe
  • 1,208
3
votes
1 answer

How to find the kernel of the homomorphism between two fundamental groups.

Let $X = T^2 - \{x_1\}$. I need to describe the kernel of the homomorphism $$\phi :\pi_1(X,x_0) \rightarrow \pi_1(T^2,x_0).$$ I know $$\pi_1(X,x_0) \cong F(a,b)$$ and $$\pi_1(T^2,x_0) \cong \pi_1(S^1,x_0) \times \pi_1(S^1,x_0) \cong \mathbb{Z}…
ABC
  • 379
3
votes
2 answers

How to calculate rigorously the degree of the map $f: S^1 \to S^1, f(z) = z^n$.

This was done in Hatcher's algebraic topology example 2.32. However I do not understand it at all. An alternative approach I know of is to show that the degree for homology matches the degree for the fundamental group which requires Hurwitz map.…
Keith
  • 1,383
3
votes
2 answers

Does two homotopic map induce the same homomorphism of homotopy groups?

Assume $f,g$ are two maps from space $F$ to $E$ homotopic, then will they induce the same homomorphism of homotopy groups? Or much stronger, assume $f$ is an inclusion from $F$ to $E$, $F$ the fiber of a fiber bundle $E$, $g$ is a constant map, then…
TheWildCat
  • 890
3
votes
1 answer

The reduced homology of wedge sums is the direct sum

I want to show that $\tilde{H_n}(\vee_{\alpha}X_{\alpha}) = \oplus \tilde{H_n}(X_{\alpha})$. Note that $\tilde{H_n}(\vee_{\alpha}X_{\alpha}) = H_n(\vee_{\alpha}X_{\alpha}, x_0)$ where $x_0$ is the basepoint. Hatcher hinted that we should consider…
koch
  • 582
3
votes
2 answers

How to compute the homology groups of spherical space forms?

Let $S^n /\Gamma$ be a spherical space form where $\Gamma
Totoro
  • 829
3
votes
1 answer

condition on existence of quillenization

$X_{Ab}$ is the Quillenization of a path-connected space $X$ if $X_{Ab}$ has abelian fundamental group, and there exists a continuous map $X\rightarrow X_{Ab}$ inducing an isomorphism $H_n(X)\rightarrow H_n(X_{Ab})$ for all $n$. Let $\pi=\pi_1(X)$.…
gmoss
  • 922
3
votes
1 answer

Using Algebraic Topology, show that $M \times N$ is orientable if and only if $M$ and $N$ are both orientable.

This is an exercise 3.3.5 in Hatcher's Algebraic Topology book. 3.3.5: Show that $M \times N$ is orientable if and only if $M$ and $N$ are both orientable. Notation: For $x \in M$, $H_n(M|x):=H_n(M,M-x)$. Definition: An orientation of an…
user625442
3
votes
1 answer

Is Projective space minus a point really homeomorphic to vector space?

In his book Fundamentals of Algebraic Topology Weintraub claims on page 96: For $k=\mathbb R$ or $\mathbb C$, $\mathbb kP^n\setminus [1,0,\cdots,0]$ is homeomorphic to $k^n$. This look fishy to me, in particular because for $k=\mathbb C, n\geq 2$…
doloreshaze
  • 49
  • 2
3
votes
1 answer

Confusion in using Mayer-Vietoris theorem

For two topological spaces $A$ and $B$, in order to show that $H(A \sqcup B) \cong H(A) \oplus H(B) $ in this question and in general I believe one can use Mayer-Vietoris to obtain the result easily. However, I don't quite understand how it applies…
user132770
  • 119
3
votes
0 answers

Relative homology: the splitting $C_*(X,A) \rightarrow C_*(X)$

Let $C_n(X)$ be the free Abelian group generated by all the singular $n$-simplices of the topological space $X$. I'm reading Bredon's Topology and Geometry and there's a statement which says: The group $C_n(X,A) = C_n(X)/C_n(A)$ is free Abelian.…
pamplemousse
  • 223
3
votes
1 answer

Degree zero singular cohomology is free?

I am considering homology and cohomology with integer coefficients. For singular homology of a topological space $X$, we know that $H_0(X)$ is free on the number of path-components of $X$, and $\widetilde H_0(X)$ (corresponding to the homology of…
user620793
  • 51
3
votes
1 answer

Homology and Mobius Band

Let $M$ be the closed Mobius band (so $M$ is a two-manifold with boundary). Suppose $U$ is an open subset of $M$ so that $\bar{U}\cap \partial M$ is empty and the inclusion $i:U\to M$ induces the zero map on homology with $\mathbb{Z}_2$…
RBega2
  • 223
3
votes
1 answer

Does semi-locally simply connected imply locally path connected?

I'm reading about the classification of covering maps, and getting a little confused about the concepts. Does semi-locally simply connected imply locally path connected?
njlieta
  • 411
3
votes
1 answer

Is the restriction of a map representing a cohomology class on its Poincare dual null-homotopic?

Let $M$ be a 5-manifold (possibly non-orientable), $g\in H^2(M,\mathbb{Z}_2)$ is represented by a map $\tilde{g}:M\to K(\mathbb{Z}_2,2)$. $\text{PD}(g)$ is the submanifold of $M$ representing the Poincare dual of $g$. $\tilde{g}|_{\text{PD}(g)}$…
Borromean
  • 641
1 2 3
…
99 100