Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
31
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
30
votes
0answers
1k views
Simplicial homology of real projective space by Mayer-Vietoris
Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n ...
14
votes
0answers
359 views
Shrinking Group Actions
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
10
votes
0answers
112 views
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 ...
9
votes
0answers
218 views
Who was Hermann Künneth?
Question as in the title:
Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia?
The well-known Künneth formula, for example in the form of ...
8
votes
0answers
84 views
Finite fundamental group in the Euclidean space
Is there an example of a (path-connected) subspace of $\mathbb{R}^3$ which has a nontrivial finite fundamental group? If not, why?
8
votes
0answers
130 views
How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?
I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
7
votes
0answers
68 views
Let $A$, $B$ be subsets of $S^n$, n≥2. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then…
Let $A$,$B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$.
I've thought to do it by contradiction ...
7
votes
0answers
117 views
When are maps between topological manifolds automatically surjective?
Take an injective (continuous) map $T:\mathbb{S}_1\to\mathbb{S}_1$. It's an obvious fact (though I can only prove it with nontrivial facts about homotopy) that $T$ is automatically surjective.
I have ...
7
votes
0answers
93 views
Lemma 6.3 of Milnor's Lectures on the h-cobordism theorem
Milnor's statement is: "Let $M^r$ and $N^s$ be sub-manifolds of $V^{r+s}$ which are all smooth, compact, oriented and without boundary. If $p$ is a point of $M^r$ contained in an $r$-cell $U$, ...
7
votes
0answers
114 views
How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?
I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be
$$\small
\dim \tilde{H}_t(X; {\mathbb{Z}}_2) = ...
7
votes
0answers
192 views
The status of $\mathbb{R}$ in homotopy theory.
The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
7
votes
0answers
2k views
deformation retract and strong deformation retract
I am trying to gain some intuition about retracts, deformation retracts and strong deformation retracts (see http://en.wikipedia.org/wiki/Deformation_retract for definitions).
We have that any strong ...
6
votes
0answers
101 views
When is a fibration a fiber bundle?
In this question I am using Wiki's definitions for fibration and fiber bundle.
I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
6
votes
0answers
86 views
When does a cohomology theory have a ring structure?
I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW ...
6
votes
0answers
86 views
Existence of a map in a Hilbert space
Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$.
Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
6
votes
0answers
85 views
Visualize Fourth Homotopy Group of $S^2$
I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
6
votes
0answers
92 views
Equivalence of two definitions of Whitehead Torsion
In their book Lecture Notes in Algebraic Topology, Davis and Kirk define the torsion of an acyclic chain complex $C$ in the following way:
Since $C$ is acyclic, there exists a simple chain complexes ...
6
votes
0answers
94 views
How to classify principal bundles over a 2 dimensional surface?
I just want to how much people know about this at the moment? I thought this is elementary and may be of execrise level, but a quick google search showed serious papers written on this subject (like ...
6
votes
0answers
80 views
A topological example from Church's undecidability paper
A. Church, in his classical paper An unsolvable problem in elementary number theory in American Journal of Mathematics Vol. 58 No. 2. (1936), pp. 345-363, (available here), wrote:
There is a class ...
6
votes
0answers
86 views
Is there a homology theory that counts connected components of a space?
It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$.
I have recently learned that for Čech homology the ...
6
votes
0answers
98 views
Poincaré duality and intersection
Let's take $X$ and $Y$ K3 surfaces and $Z\subset X\times Y$ an algebraic cycle of dimension 2. I know that the Poincarè dual of $Z$, namely $[Z]$, is in $H^4(X\times Y,\mathbb{Z})$ and by Kunneth ...
6
votes
0answers
120 views
How did Chern pictured the first Chern number?
The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
6
votes
0answers
72 views
Identification of integration on smooth chains with ordinary integration
Let $M$ be a smooth oriented $n$-dimensional manifold and denote by $A \in H_n(M;\mathbb{Z})$ the fundamental class of $M$ (a generator of singular homology consistent with the orientation of $M$). ...
6
votes
0answers
123 views
Riemann-Hurwitz Formula Using Homology
Does someone know of any good reference for a proof of the Riemann-Hurwitz Formula (of Riemann Surfaces) that uses Spectral-Sequences and Homology ?
Thanks in advance !
[ I think I know how to ...
6
votes
0answers
186 views
Cohomology ring of Grassmannians
I'm reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing):
Let $w=1+w_1+ \ldots + w_m$ be the total ...
6
votes
0answers
107 views
Finding the universal cover of a matrix group
Consider the group $G = \left\{\begin{pmatrix} a & b\\ 0 & 1\end{pmatrix}: a \in \mathbb{C}^{\times}, b \in \mathbb{C}\right\} \subset GL(2, \mathbb{C})$. How does one find the universal cover ...
6
votes
0answers
201 views
Weak Bott periodicity vs. strong Bott periodicity
Bruno Harris' proof (or I guess also Bott's original proof) of Bott periodicity (see here for instance) shows that there is a homotopy equivalence $h\colon\mathbb{Z}\times BU \rightarrow \Omega^2 ...
6
votes
0answers
140 views
Fixed Points of a Reflection
This question is about problem 2.C.5 from Allen Hatcher's Topology. The statement of the problem is as follows:
Let $M$ be a closed orientable surface embedded in $\mathbb{R}^3$ in such a way that ...
6
votes
0answers
274 views
A Way to make the following “proof” of the Hairy Ball Theorem rigorous?
I plan on giving a talk soon to undergraduates and I'd like to talk about the hairy ball theorem during the talk. I was trying to think of some sort of visually intuitive proof of this fact. (I ...
5
votes
0answers
54 views
explicit cocycle representing Stiefel-Whitney class in Milnor and Stasheff
I am trying to do Problem 7A in Characteristic Classes by Milnor and Stasheff which asks the reader to do the following:
Identify explicitly the cocycle $C^r(G_n) \cong H^r(G_n)$ which ...
5
votes
0answers
53 views
Injective Resolutions in $\mathfrak{Ab}(X)$
Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, ...
5
votes
0answers
76 views
Artin Algebraic Topology
I am reading algebraic topology by Artin, and he gives the following definition. Let $X$ be a topological space. We define a homomorphism $sd_X: S_p(X) \to S_p(X)$ by induction. If $T: \Delta_0 \to ...
5
votes
0answers
62 views
Homological definition of orientation at a boundary point?
For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
5
votes
0answers
82 views
Covering Spaces for $S^1 \vee S^1$
I'm trying to list the connected 3-sheeted covering spaces of $S^1 \vee S^1$ up to isomorphism.
I know what the answer is; I've seen listings.
My question is, if I'm deriving this, how do I know ...
5
votes
0answers
107 views
Inverse functor in proof of Dold Kan Correspondence
I´m looking at the proof of the Dold-Kan correspondence. Let $SA$ be the category of simplicial objects in an abelian category $A$ and $CH_{\ge0}(A) $ the category of non-negative chain complexes in ...
5
votes
0answers
91 views
Exercise 4-A “Characteristic Classes” by Milnor and Stasheff
Exercise 4-A of Milnor and Stasheff's book "Characteristic Classes" reads:
Show that the Stiefel-Whitney classes of a Cartesian product are given by
$w_k(\xi\times\eta) = \sum^k_{i=0} ...
5
votes
0answers
82 views
Gluing manifolds
Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this:
A point $(\cos \phi , \sin \phi, ...
5
votes
0answers
83 views
$\operatorname{Spin}^c(n)$ is a Lie group?
Assume that $n\geq 3$. (This implies $\pi_1(\operatorname{SO}(n))=\mathbb{Z}/2$.)
Let $\rho\colon \operatorname{Spin}(n)\to \operatorname{SO}(n)$ be the 2-fold universal cover. Identify $\ker(\rho)$ ...
5
votes
0answers
154 views
Examples of special sphere bundles
I'm interested in examples of sphere bundles which do not arise from vector bundles.
I'm not quite clear about the following. So please let me know if anything is false.
I believe that a ...
5
votes
0answers
62 views
question about disconnected normal covering map
I have stuck on the problem in Hatcher, Algebraic topology, which claim that if the covering map $q\circ p:X\rightarrow Y \rightarrow Z$ is normal, then the covering $p:X\rightarrow Y$ also is normal. ...
5
votes
0answers
75 views
Unicoherence of non-euclidean spaces
My question concerns the notion of unicoherence, which is a property that a topological space may or may not have. The definition (from Wikipedia) is:
"A topological space $X$ is said to be ...
5
votes
0answers
74 views
Normal subgroups of the fundamental group of a non-orientable surface.
Let $N^2_g$ be a non-orientable closed genus $g\geq 2$ surface. Is there a way to explicitly list the normal subgroups of $\pi_1(N^2_g)$ in terms of generators and relations? I am interested in ...
5
votes
0answers
74 views
diagonal image of a primitive homology class
Let $X$ be a topological space. A class $a\in H_n(X;\mathbb Z)$ is said to be primitive if $a\not = m b$ for every integer $m>1$ and $b \in H_n(X;\mathbb Z)$. Let $$\Delta_*:H_n(X;\mathbb Z)\to ...
5
votes
0answers
72 views
Continuous choice of basis for subspaces
Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
5
votes
0answers
185 views
Definition of Reshetikhin-Turaev TQFT
I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
5
votes
0answers
231 views
Homotopy extension property vs. good pairs
I'm taking a course that uses the book "algebraic topology" by Allen Hatcher. I this book there are two different ways in which a pair (X,A) of a topological space X and a subspace A can be nice: They ...
5
votes
0answers
909 views
The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent
This is exercise 1.3.8 in Hatcher:
Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then ...
5
votes
0answers
137 views
Ring structure in the Serre spectral sequence
I've tried to understand what's going on in Example 1.5 on page 27-28 in Hatcher's notes on spectral sequences. There is one part in the reasoning that I can't understand here. He writes down a table ...
5
votes
0answers
117 views
Invariance of Wall's self-intersection under the regular homotopy
For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
