Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [homology-sphere]

A homology sphere is an $n$-dimensional topological manifold that has the same integral homology as the $n$-sphere.

Let $X$ be an $n$-dimensional topological manifold. Then $X$ is a homology sphere if

$$H_k(X, \mathbb{Z}) = \begin{cases} \mathbb{Z} & k = 0, n\\\ 0 & \text{otherwise}. \end{cases}$$

The most notable example of a homology sphere is the Poincaré homology sphere which has non-trivial fundamental group. If one were to replace the simply-connected hypothesis of the Poincaré conjecture with the vanishing of degree one homology, it would no longer be true (with the Poincaré homology sphere being a counterexample).

17 questions
12
votes
1 answer

Is the universal cover of an integral homology sphere again an integral homology sphere?

Let $\Sigma$ be an integral homology sphere (from now on I will drop the word 'integral'). If $\Sigma$ is simply connected, then it is homotopy equivalent to a sphere by Whitehead's Theorem, and hence homeomorphic to a sphere by the solution of the…
Michael Albanese
  • 104,029
8
votes
0 answers

Visualizing the Poincare homology sphere

I know that past a certain point, one should graduate from the view that homology/homotopy groups "count holes" in any realistic, grounded, real-life meaning of the word "hole". However, I still want to see if there is any visual intuition to be had…
D.R.
  • 10,556
7
votes
3 answers

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group $I^*$ in view of the representation of $P_3$ as…
Rune Christiansen
  • 73
7
votes
1 answer

If $M$ is a rational homology sphere, is there a map $S^n \to M$ of non-zero degree?

Let $M$ be a closed orientable $n$-dimensional manifold. It is not hard to show that $M$ admits a degree one map $M \to S^n$ (see here for example). In fact, because $S^n$ admits maps $S^n \to S^n$ of all degrees, $M$ admits a map $M \to S^n$ of…
Michael Albanese
  • 104,029
5
votes
3 answers

Simply-connected $\mathbb{Z}_p$-homology spheres?

Let $X$ be a $\mathbb{Z}$-homology $n$-sphere, i.e., a closed manifold with $\mathbb{Z}$-homology groups of the standard $n$-dimensional sphere. If $X$ is simply-connected, it is not difficult to see (and not difficult to look up on the internet)…
user3158840
  • 177
  • 7
3
votes
1 answer

The $n$-dimensional cube modulus its boundary is homeomorphic to an $n$-dimensional sphere

Let ${I}^{n}$ denote the $n$-dimensional cube, $\partial{I}^{n}$ be its boundary and ${S}^{n}$ denote the $n$-dimensional unit sphere. Now for a pointed space $(X,x_{0})$ the $n^{th}$$homotopy$ $group$ of $X$ is : The set of homotopy classes of…
Morettin
  • 325
2
votes
1 answer

homology $n$-sphere in $\mathbb R^{n+1}$ is a sphere?

Recall that a homology sphere of dimension $n$ is an $n$-dimensional manifold $X$ for which $$ H_0(X,\mathbb Z) = H_n(X,\mathbb Z) = \mathbb Z $$ and $$H_k(X,\mathbb Z) = \{0\}, \quad k \neq 0, n.$$ Does the requirement that $X$ is embedded in…
shuhalo
  • 8,084
2
votes
0 answers

Does Alexander duality hold for compact, orientable Homology sphere?

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we rather replace $\mathbb S^n$ by a compact…
uno
  • 1,832
2
votes
1 answer

Intuition for local degree formula for singular homology map between n-spheres

In Hatcher's Algebraic Topology, he gives a formula for computing the degree of a map $f_*: H_n(S^n) \to H_n(S^n)$ in case some point $y$ in $S^n$ has preimage consisting only of finitely many points $x_i$. Then $\deg f = \sum_i \deg f \mid_{x_i}$.…
bsbb4
  • 3,751
2
votes
1 answer

Using Poincaré duality to show a closed manifold is a homology sphere

Suppose that $M$ is an orientable, compact, $(n-2)$-connected, $(2n-3)$-dimensional smooth manifold, where $n$ is a natural number. I want to show that $M$ is a homology sphere if and only if the reduced homology group $\tilde{H}_{n-1} =…
Mathmank
  • 639
  • 4
  • 14
1
vote
1 answer

Why are invariants of Homology 3-Spheres interesting?

I see how invariants (of any kind of mathematical objects) are interesting in general, since classification is interesting, but mostly in case there is a vast variety of such objects that we do not understand well (for example knots or…
Pandora
  • 6,874
1
vote
0 answers

Writing an integral over a sphere in terms of integrals of another spheres with lower dimension

Suppose $N = 4$. Given $g \in L^{1}(S^{N-1})$, I would like to know if is it possible to write $$ \int_{S^{N-1}} g(x,y,z,w) d\sigma^{N-1} = C \int_{S^{N-3}}\left( \int_{S^{N-3}} g(x,y, z,w) d \sigma^{N-3}_{(x,y)} \right) \sigma^{N-3}_{(z,w)} \qquad…
ThiagoGM
  • 1,671
1
vote
0 answers

Banach–Tarski n sphere

The Banach Tarski Paradox as applies for 3 sphere. Is there a solution for Banach–Tarski paradox for n sphere? or in particular for 5 sphere, and if so what is the general solution?
user3505444
  • 111
1
vote
1 answer

The $+$-construction on a homology $n$-sphere

I am working on Weibel's K-Book and when defining higher K-Theory for a ring via $BGL(R)^+$, I have encountered a question concerning a homology $n$-sphere. The statement I want to show is the following: Let $X$ be a homology $n$-sphere, i.e., a…
Chris
  • 453
0
votes
1 answer

"True angle" between unit vector on sphere

I have a sphere with center O = (0,0,0) and radius r=1. I want to calculate "true" angles phi(i) between points Pi lying on this sphere. From cartesian coordinate, I have : $cos(phi) = x1 x2 + y1 y2 + z1 z2$ The problem is that this angle change…
usersss
  • 11
1
2