Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
0
votes
0answers
33 views
Deformation retraction question
I was just curious as to whether the qoutient space of $\mathbb{R}^3$ obtained by the equivalence relation $x \sim -x$ deformation retracts to the $\mathbb{RP}^3$ minus a point. Thank you for your ...
1
vote
0answers
148 views
Computing singular homology
could you help me solving this questions,please
Let $D^2$ be a 2-dimensional disc and $M$ be the Möbius strip. Note that the boundary of
both $D^2$ and of $M$ is homeomorphic to the circle ...
4
votes
1answer
74 views
How we do actually compute the topological index in Atiyah-Singer?
I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing ...
0
votes
0answers
20 views
Polyhedral pair $(S^1 \times S^1 \times S^1, S^1 \times S^1 \times \{1\})$
What is the simplicial complex pair $(K_1, K_2)$ such that $(|K_1|, |K_2|)$ gives the triangulation for $(S^1 \times S^1 \times S^1, S^1 \times S^1 \times \{1\})$. Can we write down all the simplexes ...
0
votes
1answer
33 views
How to see the triangulation of an object
I am reading simplicial complex in Algebraic topology by Spanier. I read the definition of triangulation of a space and polyhedral. I cant get a picture about what we mean by triangulating a space and ...
2
votes
2answers
58 views
torus and square
How to identify Torus as the quotient of the unit square. Is there any explicit homeomorphism between the some quotient topology of the unit square and Torus. What is the quotient topology on the unit ...
1
vote
3answers
72 views
A question concerning fundamental groups and whether a map is null-homotopic.
Is it true that if $X$ and $Y$ are topological spaces, and $f:X \rightarrow Y$ is a continuous map and the induced group homomorphism $\pi_1(f):\pi_1(X) \rightarrow \pi_1(Y)$ is the trivial ...
1
vote
0answers
20 views
Covering space problem from an old Qual
Suppose that S$^1 \times P^2$
covers some space, and let $h$ be a covering
translation. Show that the induced isomorphism $h$ of $H_1(S^1, P^2)$
must be the identity
0
votes
2answers
34 views
Question about covering spaces concerning a one-sheeted covering map
I have a general question about covering spaces and covering maps. If $f$ is a one sheeted covering map of $X$, then is or is it not true that $X$ is homeomorphic to $X$?
9
votes
4answers
98 views
$S^2$ cannot cover $T$, the torus
I am trying to prove that if $N$ is a compact manifold that covers the torus $T$, then $N$ must be homeomorphic to $T$. I pretty much have the proof (used euler characteristic property and ...
2
votes
1answer
64 views
Proof for nonhomotopy
Let $Z =S^1 \times I$, and let $X = S^1 \times \{ 0 \}$ and $Y = S^1 \times \{1\}$ be two subspaces of $Z$. Let $f$ be a map from $Z$ to itself, sending $(z,t)$ to $(z \cdot e^{2 \pi it}, t)$, ...
2
votes
2answers
63 views
Finding the degree of a map
I am having trouble computing the degree of a certain map using the fact that $f: N \rightarrow M$ where $M$ and $N$ are both $n$-dimensional manifolds induces a homomorphism between the nth ...
1
vote
0answers
33 views
Group action and covering spaces
Let $X$ be a path-connected and locally path-connected topological space. The action of a topolgical group $G$ on $X$ is a covering space action. For any subgroup $H < G$, we have a composition ...
1
vote
0answers
65 views
universal abelian covering space
Let $X$ be a path-connected, locally path-connected and semilocally simply-connected topological space. (Let $(Y,p)$ be a covering space of $X$, then it is called an abelian covering space if it is ...
2
votes
1answer
86 views
Quotient space and Retractions
I'm trying to learn something about topology and category theory.
Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia)
For an ...
2
votes
3answers
76 views
Relationship between the fundamental group and the natural equivalence classes of its universal cover
For a universal covering $p: Y \to X$, under the equivalence relation $y_1 \sim y_2$ if $p(y_1) = p(y_2)$, $Y$ admits the quotient map $\, \, \, q: Y \to Y / \sim$. There is a natural bijection $\bar ...
0
votes
0answers
17 views
Understanding the topology of a variety concretely
My ultimate goal is to understand how to compute the cohomology groups of complex algebraic varieties, without having to know what a scheme is.
Therefore I want to be able to handle simple examples, ...
2
votes
1answer
35 views
Generalising a homeomorphism
Show that $\mathbb{R}^n\backslash\mathbb{R}^k$ is homeomorphic to $S^{n−k−1}× \ \mathbb{R}^{k+1}$.
I have already shown that $\mathbb{R}^2\backslash(0,0)$ is homeomorphic to $S^{1}× \ \mathbb{R}$. I ...
2
votes
0answers
55 views
Topological Manifold with ball, removed and antipodal points identified orientable?
Suppose you have a compact, orientable $(2n+1)-$manifold $M$, as in $H_{2n+1}=\mathbb{Z}$. You take a neighborhood about a point homeomorphic to $\mathbb{R}^{2n+1}$, and remove a small ball $B$. So ...
3
votes
0answers
35 views
some help on the group of unknotted
Show that the group of the unknotted $K=\{(z_z,z_2)\in \mathbb{S^3} : |z_1|=1 \}$ is infinite cyclic. where $\mathbb{S^3}$ is to be considering as the unit vectors in $\mathbb{C^2}\cong \mathbb{R^4}$.
...
2
votes
0answers
32 views
Number of sheets of Covering space of $\mathbb{T}^2$ after transformation by $SL(2,\mathbb{Z})$
Suppose you have a subgroup of $H\subset\pi_1(\mathbb{T}^2)=\mathbb{Z}\times\mathbb{Z}$, $H=\text{span}\langle u,v\rangle$. If you have an element $G\in SL(2,\mathbb{Z})$, do the covers of ...
1
vote
1answer
56 views
Some Questions on Covering maps.
I want to know a few things about covering spaces/maps. First of all, it is true that covering maps induce injective homomorphisms between fundamental groups. Is it also true that covering maps ...
1
vote
0answers
29 views
Reduced homology exact sequences
Given $H$ a homology theory, let $f: X \to P$ the unique map from $X$ to a one point space $P$. This induces a map $f_{*}: H_i(X) \to H_{i}(P)$. We define the reduced homology to be $\tilde{H}_i(X, ...
3
votes
0answers
41 views
Basic computation of a double graded spectral sequence: $^I E^0_{pq}$
Let $C=\oplus_{p\geq0, q\geq0}C_{pq}$ be a double graded group with two differentials: $d^I_{pq}:C_{pq}\rightarrow C_{p-1,q}$ and $d^{II}_{pq}:C_{pq}\rightarrow C_{p,q-1}$, with the usual assumption ...
1
vote
0answers
24 views
Cohomology ring of $\mathbb R P^\infty$ with $\mathbb Z_{2k}$ coefficients
Let $k$ be a positive integer. I am trying to show that as rings, $H^*(\mathbb RP^\infty ; \mathbb Z_{2k}) \cong \mathbb Z_{2k}[a,b]/(2a , 2b , a^2 - kb)$. This is exercise 3.2.5 in Hatcher. The ...
1
vote
1answer
61 views
torus by identifying two equivalent points (mod $\mathbb{Z^2}$)
How to visualize the quotient space $\mathbb{R^2}/ \mathbb{Z^2}$ to be a torus?
you may also refer me to some books or websites. Because I want to see how the knot torus winds in this case.
thank ...
2
votes
1answer
49 views
How to show that the fundamental group of a based space is trivial.
Is there a sort of general method of proving that the fundamental group of a based space is trivial? I am trying to understand how to prove that a space is simply connected and I am not 100% sure of ...
5
votes
1answer
83 views
de Rham comologies of the $n$-torus
I'm attempting to calculate the de Rham cohomologies of the $n$-torus: $n \choose k$.
I'd like to use a Mayer-Vietoris sequence relating $H^kT^{n}$ to $H^kT^{n-1}$ and $H^{k-1}T^{n-1}$ so I can use ...
2
votes
2answers
83 views
Homology of connected sums
How do you compute the homology groups of $n$ connected sums $H_m(T^2\#T^2\#\dots \#T^2)$ where $T^2$ is the cross product of 2 circles? I know how to compute the homology groups of $T^2$ minus a ...
2
votes
0answers
39 views
Simplicial cup product on torus
I'm trying to compute the simplicial cup product on the torus (using $\Delta$-complexes) but running into a problem: each way I draw the fundamental polygon I get different answers! When I draw it as ...
3
votes
1answer
68 views
What do elements of the first homology group mean topologically?
By Hurewicz theorem we know that $H_1(X)$ is the abelianization of $\pi_1(X)$. Let $X = S_1 \vee S_1$. Mark the the loops ($S_1$) by $a$ and $b$. Then $\pi_1(X) = \langle a,b\mid \;\rangle$ which is a ...
0
votes
2answers
33 views
Continuity with subspaces
Let $f:X\rightarrow Y$ be a continuous between topological spaces. Show that the restriction of $f$ to any subspace $A \subseteq X$ is continuous.
I am studying for an exam and the answer to this ...
2
votes
1answer
48 views
Simplicial cohomology of $ \Bbb{R}\text{P}^2$
I've managed to confuse myself on a simple cohomology calculation. I'm working with the usual $\Delta$-complex on $X = \mathbf{R}\mathbf{P}^2$ and I've computed the complex as ...
1
vote
1answer
43 views
Induced homomorphisms in a universal covering map
I don't understand what happens in the induced homomorphism of a universal covering map.
A covering map $p:\tilde X \to X$ is universal if $\tilde X$ is simply connected, so the fundamental group ...
5
votes
0answers
60 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 ...
2
votes
2answers
73 views
Universal Cover of a space
I do not know what tools one uses to find the universal cover of a space. In particular I want to find the universal cover of two copies of $RP^3$ glued together at a single point at the endpoints by ...
0
votes
0answers
22 views
Let $X_\alpha$ be the connected components by arcs of $X$ (homology)
Let $X_{\alpha}$ the connected components by arcs of $X$ if $A\subset X$ and $A_{\alpha}=A\cap X_{\alpha}$ Then $H_{\ast}(X,A)=\bigoplus H_{\ast}(X_{\alpha},A_{\alpha})$
1
vote
1answer
45 views
connected sum of torus with projective plane
I would like to understand how to prove that the connected sum $\mathbb{R}P^2 \# T^2$ of the projective plane with a torus is homeomoprhic to $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$.
I got ...
3
votes
1answer
41 views
On the homotopy type of unions of 2-spheres
Here is the problem I am stuck on: If $X$ is a connected Hausdorff space that is a union of a finite number of $2$-spheres, any two of which intersect in at most one point, then show that $X$ is ...
2
votes
1answer
48 views
Different possibilities defining $\eta^2$ in the ring of stable homotopy groups?
The Hopf fibration $\eta:S^3\to S^2$ represents the generator of the first stable homotopy group $\pi_1^s$.
The direct sum of the stable homotopy groups
$$
\pi_*^s=\bigoplus_{k\in\mathbb{Z}} \pi_k^s
...
0
votes
1answer
28 views
Question about the fundamental group of simplicial complex.
Suppose we have a simplicial complex G that is finite connected.
(1)The fundamental group of G is finite;
(2)The universal cover of G is compact.
Question:
Can (1) implies (2)?
Thanks.
2
votes
0answers
44 views
Very Simple Universal Covering problem
A space $X$ is constructed from two disjoint copies of $RP^3$ and a copy
of the unit interval $I$ by gluing one end of $I$ to a point of one copy of
$RP^3$, and gluing the other end of $I$ to the ...
1
vote
1answer
59 views
Algebraic Topology Question
I was working through some old qualifying exam problems and I am struggling with this one. Any help would be greatly appreciated.
Let $n$ be a non-negative integer. For which values of $k = 0, ..., ...
3
votes
2answers
69 views
Attaching a cell
Could you help me to explain this argument:
Let $f: S^{n-1} \rightarrow A$ for $n \ge 1$, form
$$X= C(f) := \dfrac{A\coprod D^n}{f(x) \sim x, \forall x \in S^{n-1}}$$
"$(D^n,S^{n-1}) \rightarrow ...
0
votes
1answer
43 views
Question on null-homotopic maps
I am not sure how to start the following question. Any help will be greatly appreciated! Thank you!!!
Let $K$ be a 3-dimensional simplicial complex, and let $f : K \rightarrow \mathbb{R}P^2 \times ...
2
votes
0answers
30 views
Covering map Problem and induced isomorphism
I don't know how to start the following question. Any help will be appreciated! Thank you!
Suppose that $S^1 \times P^2$ covers some space, and let $h$ be a covering
translation. Show that the ...
2
votes
1answer
38 views
What map of ring spectra corresponds to a product in cohomology, especially the $\cup$-product.
Let $E$ be a spectrum, $X$ a CW-complex and associate a graded abelian group
$$
E^*(X)=\bigoplus_{k\in\mathbb{Z}}[X_+,S^k\wedge E]
$$
to it. The brackets denote stable homotopy classes. Please let me ...
0
votes
1answer
19 views
Is connectivity of a spectrum $E$ representing a cohomology theory equivalent to $E^{>1}(*)=0$?
Let $E$ be a spectrum, $X$ a CW-complex and set
$$
E^k(X)=[X_+,S^k\wedge E]
$$
for an integer $k$ as the $k$-th cohomology group of $X$ associated to the spectrum where the brackets denote stable ...
1
vote
1answer
56 views
fundamental group of a torus
In finding the fundamental group of the torus by using the Van Kampen theorem, I was reading this: "Let us choose the covering consisting of the
punctured torus U, an open disk V that covers the ...
3
votes
2answers
110 views
Orientability of projective space
Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even.
First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with ...
