Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
3
votes
1answer
19 views
Cohomology $H^ {i}(\mathbb{R}P^n, \mathbb{Z}_2)$
I have to compute $H^{i}(\mathbb{R}P^2,\mathbb{Z}_2)$. I know that is $\mathbb{Z}_2$ for $i=0,1,2$ but I'm looking for a proof without universal coefficient theorem. Have you some ideas?
2
votes
1answer
42 views
What's the fundamental group of $E^2\setminus Q^2$
Here $E^2$ is the two-dimensional Euclid space and $Q$ is the set of all rational numbers. Regard $E^2\setminus Q^2$ as a subspace of $E^2$. So what's its fundamental group and how to represent it? I ...
1
vote
1answer
23 views
Torus interior homeomorphic to torus exterior
Let $T^2 \subset \mathbb{R^3}$, then $X_i$ be its interior and $X_e$ its exterior. By computing homotopy groups of $X_i \cup T^2$ and $X_e \cup T^2$ and corresponding isomorphisms between, one could ...
2
votes
2answers
45 views
Show that $[l_1 \cdot l_2 \cdot l_3 ] = [l_1 + l_2 + l_3] \in H_1(X)$ The first Homology group of X
Let $l_1$ , $l_2$ and $l_3$ be three paths in X with $l_1 (0) =
l_3 (1)$, $l_1 (1) = l_2 (0)$ and $l_2 (1) = l_3 (0)$. Define the loop $l = l_1 \cdot l_2 \cdot l_3 $ (based at $l_1 (0)$).
Show that ...
1
vote
0answers
45 views
Compactness property
Let $\Omega \subset X$, X: Banach space. Given $\varepsilon \ge 0$, we define the set of $\varepsilon-normals$ to $\Omega$ at $\bar{x}$$\in \Omega$ by:$\widehat N_\varepsilon(\bar ...
2
votes
1answer
51 views
Infinite products of a (finite) group
So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
2
votes
0answers
40 views
$\pi_1$ and $H_1$ of Symmetric Product of surfaces
Let $X=Sym^d(\Sigma_g)$ be the d-fold symmetric product of a genus-g surface, $d\ge 2$.
Is there / what is a (quick simple) way to see that $\pi_1(X)$ is abelian?
The link in the comments ...
2
votes
1answer
50 views
question from hatcher basic 3 manifolds
The question is: why should a homologically trivial embedded sphere in a simply connected (not necessarily compact) 3 manifold M bound a compact 3 manifold embedded in M?
I had this problem reading ...
7
votes
3answers
123 views
Homology of $S^1 \times (S^1 \vee S^1)$
I'm trying to solve question 2.2.9(b) in Hatcher's Algebraic Topology.
Question: Calculate the homology groups of $X=S^1 \times (S^1 \vee S^1)$.
My attempt: I try to use the Mayer-Vietoris ...
4
votes
1answer
47 views
Computing Chern Classes of Tautological Line Bundles
I cannot find any references with how to handle this tautological line bundle $V \rightarrow P(\mathbb{H}^2)$ where $\mathbb{H}$ are the Hamilton quaternions.
I know it is a complex rank 2 vector ...
0
votes
0answers
16 views
Mixture spaces with non empty algebraic interior
I am looking for examples of mixture spaces with non empty algebraic interior where a mixture space satisfies the following properties:
In particular M
is endowed with a mixing operation ...
0
votes
0answers
26 views
K and KO spectra
In Switzer's algebraic topology book, ch 11 page 216, he defines the K and KO spectra. He then goes on to say: "Since they are $\Omega$-spectra, we have
$\tilde{KO}^0(X) \cong [X,x_0;\mathbb{Z} ...
1
vote
1answer
72 views
Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$?
Is $\def\Aut{\operatorname{Aut}}\Aut(\mathbb{I})$ isomorphic to $\Aut(\mathbb{I}^2)$ ? ($\mathbb{I},\mathbb{I}^2$ have their usual meaning as objects in $\mathsf{Top}$).
I show some of one of my ...
1
vote
2answers
71 views
Is the winding number of a map $\Omega: \; S^n \mapsto S^n$ dependent on the radius of both spheres?
I have 2 questions regarding homotopy groups.
My first questions comes from the fact that I've read different definitions of homotopy groups. The book by Manton and Sutcliffe defines them as:
...
1
vote
1answer
54 views
filtration on the (co)homology of a space from the filtration of a space
Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let
...
10
votes
1answer
99 views
Functoriality of the Fundamental group
The fundamental group is a functor from the category of pointed topological spaces to the category of groups.
Therefore every base-point preserving continuous function $f$ between pointed ...
3
votes
0answers
41 views
What is the geometric meaning of powers of the first Stiefel-Whitney class?
If $M$ is an $n$-dimensional manifold, I know $\\w_1^n(M)$ is a Stiefel-Whitney number of the manifold, but what does its vanishing geometrically mean?
More generally, does the Stiefel-Whitney height ...
0
votes
1answer
56 views
E measurable with m(E) < $\infty$?
Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$.
ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable.
I told my ...
9
votes
1answer
108 views
Why base point makes a huge difference?
While preparing a talk, I was tempted to "prove" the following relationship:
$$\text{Prin}_{G}(X)\cong [X,BG]\cong [B[\pi_{1}(X),BG]\cong [\Omega B[\pi_{1}(X)],\Omega BG]\cong [\pi_{1}(X),G]$$
Here ...
1
vote
3answers
57 views
Disc with two points identified
Is a disc $D^2$ with two points on the boundary identified, same as $D^2 \vee D^2$ ? They both have boundary $S^1 \vee S^1$. I am confused because an exercise in Hatcher seems to ask the same question ...
-1
votes
1answer
52 views
Isomorphism $\left(\mathbb{C}^{n}\setminus\{0\}\right)/\mathbb{Z}$ with $S^{1} \times S^{2n-1}$
I have to prove that there is this isomorphism:
$$\frac{\mathbb{C}^{n}{\setminus\{0\}}}{ \mathbb{Z}} \simeq S^{1} \times S^{2n-1},$$
where there is this equivalence relation in the left side:
$(w_1, ...
0
votes
0answers
21 views
Need an application of Morse theory for second-order differentialle systems
I'm looking for some applications of Morse theory for the second order differentialle systems,
Someone can help me with a pdf or a book or an article which has these applications ?
Please
Thank ...
1
vote
1answer
30 views
Uniqueness of Seifert graphs
If we make the bands and disks of a Seifert surface really small and really thin the surface collapses to a graph. It is called a Seifert graph.
If it is not a directed and weighted graph, can we ...
1
vote
2answers
40 views
Complex Solutions to Polynomials
I'm trying to use topology to prove that:
$z^{n} + a_{n-1}z^{n-1} + ... + a_{1}z + a_{0} = 0$
has a solution in $\mathbb{C}$ if and only if, for each positive real number $c$, the equation
$z^n + ...
3
votes
0answers
49 views
Tangent bundle of Grassmann manifold
I have to prove that the tangent bundle of Grassmann manifold $G_n(\mathbb{R}^{n+h})$ is isomorphic to $\operatorname{Hom}(\gamma^n(\mathbb{R}^{n+k}),\gamma^\perp)$, with $\gamma^{\perp}$ is the ...
0
votes
1answer
45 views
A map on the $2n$-sphere and degree zero
Let $f : \mathbb{S}^{2n} \to \mathbb{S}^{2n}$ is continue e $f(x) \neq -f(-x) \quad\forall x$. Prove that $f$ has degree zero.
Thanks!!
2
votes
0answers
77 views
Imagining four or higher dimensions and the difference to imagining three dimensions
I’m very interested in how people envision four or higher dimensions.
And I’m especially interested in how geometers and topologists who actually work in four dimensions do.
Now I know of the video ...
4
votes
1answer
47 views
studying the topology of a real algebraic set
Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
3
votes
2answers
86 views
Homology of Möbius Strip
How does one calculate the homology groups of the Möbius strip?
I'm thinking of two methods.
Use cellular homology. I tried to draw a delta-complex structure of the Möbius strip but I'm not sure if ...
-6
votes
0answers
68 views
Are there two or three unique paths for tonal movement in music? [closed]
In music theory we learn there are 2 unique tone value class pathways that include in sequence every note in an octave: the chromatic path and the circle of 5ths. This makes a toroidal manifold. We ...
11
votes
1answer
102 views
If $S^1 \hookrightarrow X$ induces an injection of $H_1$, then $X$ retracts onto $S^1$
This is an exercise in Hatcher, section 4.3, exercise 3, page 419, on which I'm struggling.
Suppose that a CW complex X contains a subcomplex $S^1$ such that the inclusion
$S^1 \hookrightarrow ...
3
votes
1answer
56 views
Spectral sequences: equivalence of exact couples and classic (?) method
By the 'classic' method I mean the construction of the spectral sequence associated to a filtration as found in Weibel's book p. 133-134. There is also the method of construction through exact couples ...
1
vote
1answer
40 views
Graphs from Seifert surfaces
Given a Seifert surface if we make the disks and bands infinitely small and thin it becomes a graph where the disks are vertices and the bands are edges. Can we say that following theorem,
For ...
1
vote
2answers
42 views
example of homotopy which is not path homotopy
Can someone give me a simple, concrete example of a homotopy, which is not a path homotopy?
Let $f, f'$ be continuous maps from $X$ to $Y$, and let $F: X \times I\to Y$ a continuous map such ...
2
votes
0answers
49 views
The closed orientable surface of genus 2
I have a very simple question to ask. What is a closed orientable surface of genus 2?
Thank you in advance for helping me.
-1
votes
1answer
56 views
Does $S^1 \times S^2$ nontrivially covers itself?
I am not sure how to show whether or not $S^1 \times S^2$ nontrivially covers itself. Some help would be appreciated.
Thanks
2
votes
1answer
71 views
Does the sphere $S^3$ nontrivially cover itself?
I am having a hard time deciding whether or not $S^3$ nontrivially covers itself. Some help would be appreciated.
Thanks
2
votes
0answers
31 views
Determining a preimage of the connecting homomorphism of the Mayer Vietoris sequence
In this article http://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence, and in the section about the connecting homomorphism associated to a Mayer-Vietoris sequence, the connecting ...
3
votes
1answer
35 views
Sufficient condition for a direct limit of abelian groups to be infinitely generated
I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
0
votes
1answer
27 views
Chern classes tangent bundle $\mathbb{C}P^n$.
Let $V \in Vect_k(M, \mathbb{C})$. We define Chern classes $c_i(V) \in H^{2i}(M, \mathbb{Z})$ with the usual 4 axioms. Now we consider the tangent bundle
$$ \mathbb{C}^n \hookrightarrow T\mathbb{C}P^n ...
1
vote
2answers
46 views
Uniqueness of Seifert surfaces of knots
I know the theorem that Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. So, can we also say that for every unique Seifert surface there is an unique knot and vice ...
9
votes
1answer
83 views
Homology of $\mathbb{R}^2$ under the equivalence $x \sim 2x$
I was computing some examples of homologies of quotient spaces and I thought of the following. Does anyone know how to compute the homology groups of $\mathbb R^2/\sim$, where $\sim$ is the ...
4
votes
1answer
44 views
Constructing a odd homeomorphism between $A$ and $S^n$.
Let $A\subset\mathbb{R}^N\setminus\{0\}$ be a closed symmetric set ($x\in A$ then $-x\in A$). Suppose that $A$ is homeomorphic to some sphere $S^n$, $n\leq N$ ($n$ is the dimension of the sphere). Is ...
0
votes
1answer
98 views
number of simplices in barycentric subdivision
Let $K$ be a simplicial complex. Is there a way to calculate the number of k-simlices in the barycentric subdivision $K'$ of $K$? Given the number of $l$-simplices in $K$, for any $l$, of course.
(I ...
2
votes
1answer
51 views
Need help understanding statement of Van Kampen's Theorem and using it to compute the fundamental group of Projective Plane
I'm using the statement from Hatcher. I really don't understand the statement of the theorem, let alone the proof, and I especially don't understand what the normal subgroup $N$ generated by ...
1
vote
1answer
27 views
Action of fundamental group on n-th homotopy groups for RP^n
Is there any short way to see that the action of $\pi_{1}(RP^{n})$ on $\pi_{n}(RP^{n}) = \mathbb{Z}$ is trivial for $n$ odd and nontrivial for $n$ even?
Maybe something without much machinery (smth ...
2
votes
1answer
61 views
How to compute the homology group $H_q(\mathbb{R}^n-e^r)$?
Let $e^r$ be a homeomorphic copy of $I^r$ in $\mathbb{R}^n$($I=[0,1]$).How to compute the homology group $H_q(\mathbb{R}^n-e^r)$?($r,n,q$ are non-negative integers)
2
votes
0answers
135 views
Lemma of Whitehead
this is the lemma of Whitehead
And i really don't understand the proof
How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ )
and how to ...
4
votes
2answers
55 views
Looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex
I am looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when $n=2$. ...
1
vote
0answers
33 views
Does group of deck tranformations acts transitively on each fibre if it acts traansitively on one fiber?
i am reading bredon "Topology and Geometry "
It states that if we have a covering map p : X ->Y s.t. p(x) = y.X,Y are Hausdorff, path connected and locally path connected etc.
I have 2 questions:
...
