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
6
votes
2 answers

Proof of the Ham-Sandwich theorem

I have doubts about the proof of the Ham-Sandwich theorem descibed on planetmath (http://planetmath.org/proofofhamsandwichtheorem) and wikipedia (http://en.wikipedia.org/wiki/Ham_sandwich_theorem): There you fix one of the $n$ sets in $\mathbb R^n$…
Jochen
  • 12,254
6
votes
2 answers

Does every continuous map induce a homomorphism on fundamental groups?

Let $X$, $Y$ be topological spaces and $f:X \to Y$ be a continuous map. Does $f$ induce a homomorphism $f_* : \pi_1(X) \to \pi_1(Y)$? If not, what are the conditions on $f$ so that $f_*$ would be a homomorphism? My motivation for knowing this is an…
Dávid Natingga
  • 2,889
  • 2
  • 27
  • 41
6
votes
3 answers

CW complex that deformation retracts to annulus and Möbius band

I am trying to construct a 2-dimensional CW complex that contains both an annulus, thought of as $S_{1}\times I$ and a Möbius band as deformation retracts. The first part of the problem asked to show that the mapping cylinder of every map…
cyc
  • 2,973
6
votes
2 answers

Lefschetz number equal to Euler Characteristic of Fixed Points

If $X$ is a finite simplicial complex and $f:X\rightarrow X$ is a simplicial homeomorphism, show that the Lefschetz number $\tau(f)$ equals the Euler characteristic of the set of fixed points of $f$. My progress: We know the set of fixed points…
ergo
  • 161
6
votes
1 answer

Reduced homology group of wedge sum

Let $X$ be Hausdorff space and let $x \in X$ such that there is a closed neighborhood $U$ such that $\{x\}$ is a strong deformation retract of $U$. If $Y$ is another Hausdorff space, I would like to show the inclusion maps induce…
user73699
  • 61
6
votes
3 answers

Reduced Homology on unreduced suspension

I want to establish the result for any homology theory, there is a natural isomorphism which gets us $\tilde{H}_{p+1}(\Sigma X) \cong \tilde{H}_p(X)$. Most places use the Mayer-Vietoris sequence to get this, but since I haven't covered that yet, I…
user71500
  • 61
6
votes
1 answer

Using Hurewicz to approximate a space while just using its homology groups.

Let $X$ be a simply connected space with $H_2(X; \mathbb{Z}) \cong \mathbb{Z}/5$, $H_3(X; \mathbb{Z}) \cong \mathbb{Z}$ and all higher homology groups zero. Show there exists a CW-complex $Z$ with one $0$-cell, one $2$-cell and two $3$-cells and a…
user388557
  • 2,544
6
votes
1 answer

Action of fundamental group on a fiber

Let $p:(E, e_0)\to (X, x_0)$ be a covering. If $\alpha:[0, 1]\to X$ is a path beggining at $x_0$ and $e\in p^{-1}(x_0)$, then we can uniquely lift $\alpha$ to a path $\alpha'_e:[0, 1]\to E$ beggining at $e$, such that $p\alpha'_e =…
Jakobian
  • 10,247
6
votes
1 answer

Map $f: S^n \to S^n$ that extends over $D^{n+1}$ must identify a pair of antipodal points

There is a problem in Armstrong's Basic Topology in chapter 9.3 that I just can't seem to solve by myself, neither have I found a solution online. It goes as follows: Suppose we have a map $f:S^n \to S^n$ that extends over $D^{n+1}$ in the sence…
Itoro Jóska
  • 95
6
votes
1 answer

Parallelizable spheres and whitney classes of bundles

In Milnor's paper, He asserts the following corollary numbered 2 If $\mathbb{R}^n$ has bilinear product with no zero divisors, then $n=1,2,4$ or $8$. In his proof he constructs a vector bundle $E$ on $S^n$ by clutching map $$x \mapsto (\text{left…
Mahmoud Abdelrazek
  • 478
  • 3
  • 10
6
votes
1 answer

How do I figure out the coproduct on graded algebras?

I have to figure out the duals to a couple of graded algebras. This requires a comultiplication (also called a coproduct in Hatcher). Hatcher's book shows what form the comultiplication must take using a cohomology argument for the cohomology of…
Sven
  • 769
6
votes
1 answer

Torsion or Non-Torsion subgroup of $H_i$ do not define a homology theory.

I need to show that if we let $T_n(X,A)$ denote the torsion subgroup of $H_n(X,A)$, then the functor $(X,A)\mapsto T_n(X,A)$ with the obvious induced homomorphisms and boundary maps do not define a homology theory. Using the axioms that Hatcher…
user9402
6
votes
0 answers

Bing's house, but with n rooms?

In Chapter 0 of his Algebraic Topology, Hatcher presents Bing's house with two rooms as an example of a space that is contractible, but "not in any obvious way." He then asks, as an exercise (Chapter 0, exercise 8), for the reader to construct a …
Cullen Schaffer
  • 375
6
votes
2 answers

$\mathbb{R}^2$ with $n$ points removed is homotopy equivalent to the wedge sum of $n$ circles.

Let $E\subseteq\mathbb{R}^2$ consist of $k$ points. Show that $\mathbb{R}^2\setminus E\simeq\bigvee_1^kS^1$ (homotopy equivalent). I can see this geometrically, but is there a rigorous way to prove that? At the bottom of page 54 from Tammo tom…
Yuxiao Xie
  • 8,536
6
votes
1 answer

$S^1×[0,1]$ to $S^1×[0,1]$ fixed point?

Let $f : S^1 × [0,1] → S^1 × [0,1]$ denote a null homotopic map. Show that f has a fixed point. (hint: does f lift to a covering space?) I know f may lift to $\mathbb{R}×[0,1]$ (Or not?), but how to proceed to prove it has a fixed point? Is this…
Gnon
  • 73
1 2 3
…
99 100