Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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7
votes
2answers
922 views

how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
5
votes
1answer
59 views

Covering spaces of $\mathbb{RP}^n\times\mathbb{RP}^n$ for $n > 1$.

The fundamental group of $X = \mathbb{RP}^n\times\mathbb{RP}^n$ is just $G=\mathbb{Z}_2\times \mathbb{Z}_2$ when $n > 1$. So connected coverings of $X$ correspond to subgroups of $G$. This has $5$ ...
4
votes
2answers
65 views

$\pi_n(SU(2)/Z_N)\simeq?$, $\pi_n(SO(3)/Z_N)\simeq?$, $\pi_n(U(1)/Z_N)\simeq?$

So based on the tool, I have attempted to compute the following threes homotopy groups. $\pi_n(SU(2)/Z_N)\simeq?$ $\pi_n(SO(3)/Z_N)\simeq?$ $\pi_n(U(1)/Z_N)\simeq?$ With a fixed positive integer ...
4
votes
3answers
216 views

Application of Hodge decomposition

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. ...
0
votes
0answers
20 views

A Question about base change in a Galois extension

Let $K$ be a field of characteristic $0$, $K(c_1,\dots,c_n)$ be the rational functional field of $n$ indeterminates. Consider the splitting field of ...
3
votes
2answers
277 views

Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell

Prove: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ and $X\sqcup_gD^n$ are homotopy equivalent. I think it can be proved by showing they are both deformation retracts of ...
2
votes
1answer
35 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
0
votes
1answer
79 views

Relationship between fundamental polygon and its side edges

Here is the fundamental polygon diagram for torus and the diagram for its edge of the square region: My question is why the direction of loops in both circles in the right diagram must be ...
0
votes
2answers
61 views

Isometry of surfaces in $\mathbb{R}^3$

Let $F$ be an isometry of the Euclidean space $\mathbb{R}^3$. Hence $F$ is orthogonoal transform followed by translation by a constant vector. Let M be a surface of $\mathbb{R}^3$ that is connected, ...
2
votes
0answers
93 views

Conditions for a Topological space to be a Spectrum

I'm looking for conditions for a topological space $X$ to be a Spectrum. A topological space $X$ is a spectrum if it can be delooped infinitely (more accurately, «double-infinitely»). Some ...
1
vote
1answer
71 views

Aplanar covering of $S^1 \vee S^1$?

Can someone provide an aplanar covering of $S^1 \vee S^1$? What if I insist on it being finite degree? (This question is motivated by the diagram in the first chapter of Hatcher's Algebriac Topology, ...
2
votes
1answer
105 views

Algebra prerequisites for Homology Theory

I am a first year graduate student in Mathematics. I am planning to take a graduate course on Homology Theory. My background is Point Set Topology (material covered in Part 1 of Munkres) and the ...
2
votes
1answer
74 views

About existence of Morse functions

Let's consider 4-manifold $M$, $\partial M = \partial M_1 + \partial M_2 = S^1 \times S^2 + \mathbb{RP}^3$. Is it true that there exist a Morse function $$f\colon M^4 \to [0,1],\quad f^{-1}(0) = ...
2
votes
0answers
38 views

Knowing the existence of a fixed point set from an induced fundamental group automorphism

Let $L$ be a link in $S^{3} $ and $f_{ \phi } : \pi_{1} (S^{3} \backslash L ) \rightarrow \pi_{1} (S^{3} \backslash L )$ be induced from a periodic map $\phi $ of $S^{3} $, restricted to the ...
3
votes
1answer
123 views

Cohomology groups

I have some questions. 1) I tried to compute the cohomology group of $S^3$ with coefficients in $\mathbb{Z}/2\mathbb{Z}$ but I don't know if my result $$ H^k(S^3,\mathbb{Z}/2\mathbb{Z}) = ...
3
votes
1answer
150 views

Is a simply connected set connected?

A set $X$ is considered connected if there is no separation of the set $X$ into disjoint sets $A,B$ such that $X = A \cup B$, where neither sets ($A$,$B$) contain limit points of each other. Now a ...
3
votes
2answers
58 views

When is a regular map a covering map?

Let $M$, $N$ be two manifolds of the same dimension. A map from $M$ to $N$ is regular provided its tangent map is one to one. A map from $M$ to $N$ is a covering map provided each point in $N$ has a ...
2
votes
1answer
81 views

CW-complex with zero boundary operators

If I have a CW-complex, is it possible to find a homotopically equivalent one that will have zero boundary operators? It shouldn't be always possible to find such a triangulation for the initial ...
1
vote
0answers
85 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
2
votes
1answer
61 views

For which $n$, is any continuous map from $S^n$ to $S^1 \times S^1$ nulhomotopic.

For which $n$, where $n$ is a positive integer, is any continuous map from $S^n$ to $S^1 \times S^1$ nulhomotopic. If every continuous map from some $S^n$ to $S^1 \times S^1$ was nulhomotopic, would ...
6
votes
4answers
218 views

Simply Connected domains.

If $U$ and $U'$ be two domains in $\Bbb C$, and $f$ be a homeomorphism in $U$ and $U'$ then domain $U$ is simply connected $\iff$ $U'$ is simply connected. I found this problem in complex analysis. So ...
1
vote
1answer
44 views

Generators of the first singular homology group of a Riemann surface

Let $X$ be a Riemann surface embedded in $\mathbb C^2$ with coordinates $(z,w)$ and let $\pi_z \colon X \to \mathbb C$ be the projection on first coordinate with property that that for cofinite number ...
2
votes
2answers
74 views

Fundamental group of the surface of a cube with interior of all edges removed

Find the fundamental group of the surface of a cube with interior of all edges removed (i.e. the space which consists of the vertices and interior of the faces of the cube. Can I deformation retract ...
1
vote
2answers
146 views

Cohomology groups for the following pair $(X,A)$

Let $X=S^1\times D^2$, and let $A=\{(z^k,z)\mid z\in S^1\}\subset X$. Calculate the groups and homomorphisms in the cohomology of the exact sequence of the pair $(X,A)$. I know that theorically one ...
4
votes
1answer
36 views

Nonexistence of map taking boundary to torus to wedge of circles homeomorphically

Specifically, the question says to consider the torus $T$ as a square with the usual identifications, with two opposite boundary edges labelled $a$ and the other two edges labelled $b$, and consider ...
4
votes
2answers
182 views

Find the fundamental group of the product of two circles with the diagonal removed.

Let $X = S^1 \times S^1 - \Delta$ , where $\Delta = \{ (x,y) \in S^1 \times S^1 | x = y \}$. Determine the fundamental group of $X$. I know $\pi_1 (S^1 \times S^1) = \pi_1 (S^1) \times \pi_1 (S^1) = ...
1
vote
1answer
59 views

Abelianized fundamental group

Let $P$ be the projective plane and let $nP$ be the connected sum of $n$ copies of the projective plane. Show that the abelianized fundamental group $\pi_{1}(nP)/[\pi_1,\pi_1]$ is the direct sum of a ...
2
votes
1answer
102 views

Antipodal points of sphere

Whenever $S^2$ is the union of three closed subsets $A_1$, $A_2$, and $A_3$, then at least one of these sets must contain a pair of antipodal points {${x,-x}$} in $S^{2}$ This is homework from ...
3
votes
2answers
203 views

Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre ...
1
vote
2answers
151 views

Are spaces with isomorphic fundamental groups homotopically equivalent?

I know that the converse of this statement is true but I am not sure how to go about finding out the answer to this question.
1
vote
1answer
134 views

Quaternion Projective Space

So $\mathbb{R}\mathbb{P}^n$ and $\mathbb{C}\mathbb{P}^n$ can be built as CW complexes. Is there an analogous construction if we consider over the quaternions? How about the octonions? Can we give ...
2
votes
0answers
41 views

Covering Spaces and Fundamental Groups

Can somebody tell me if what I did is right? I need to Draw the based cover $\hat{B}\rightarrow B$ such that $\pi_{1}(\hat{B},v)$ corresponds to the subgroup $\langle a^{3}, a^{2}b\rangle$ ...
0
votes
1answer
36 views

Correspondence between set of paths

Let $x,y \in X$. Denote by $P(x,y)$ the set of equivalence classes of paths in $X$ from $x$ to $y$ under the relation: Homotopic equivalence relative to $\{0,1\}$. Then Can we say that $\exists$ a ...
1
vote
2answers
49 views

Trying to understand difference between loops/paths and functions

I'm learning the very basics of algebraic topology right now and one thing has got me really confused. For example, take $S^1$ the circle. Then we know that a loop around the circle once given by ...
1
vote
1answer
95 views

Zero Cohomology means zero homology?

Suppose we have a space $X$, which has zero cohomology (except in degree zero). Does he neccesarly have zero homology (except in degree zero)? If not, what if $X$ is a manifold? Universal ...
2
votes
1answer
55 views

need help with problem on homology group

Let $A_n=\{z\in \mathbb{C}\mid z^n$ is non-negative real number$\}$ then find $H_1(A_n,A_n-\{0\})$ $H_1(A_n,A_n-\{z\})$ when $0\not=z\in A_n$ show that $A_n$ is not homeomorphic to $A_m$ when ...
2
votes
1answer
145 views

Union of 2-sphere with line segment in $\mathbb{R}^3$ removing one point homotopy equivalence.

I am working on a problem from Lee's Introduction to Topological Manifolds where one is asked to compute fundamental groups using Van Kampen's theorem. I know how to use Van-Kampen's theorem but I ...
3
votes
2answers
81 views

Prove these 3 spaces are homotopy equivalent

The image is below. (a) $S^2$ with a diameter. (b) $T^2$ with a disk in the middle hole. (c) $S^2$ tangent with $S^1$ . I think they may the deformation retract of the same space. But I can't ...
2
votes
0answers
69 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
2
votes
1answer
83 views

Loop is contractible iff it extends to a map of disk

Let $X$ be a space, $f : S^1 \to X$ be continuous function. $f$ is homotopic to constant map $h = c$ iff $\exists$ continuous $g: D^2 \to X $ such that $g |_{S^1} = f $ My Attempt Take a homotopy ...
2
votes
0answers
103 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
1
vote
1answer
46 views

Invariant homology classes

Let $G$ be a finite group acting freely on a manifold $X$. What is the geometrical meaning of invariant homology classes $H_i(X,\mathbb Z)^G$? The same question for coinvariants $H_i(X,\mathbb Z)_G$.
2
votes
0answers
33 views

Degree and picture of a Covering map

I need to know if I am right: I need to know the degree of this covering map $R \rightarrow S$: $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ I have that genus of $R$, $g_{R} ...
0
votes
0answers
27 views

Affine operations on affine simplices

First of all some definitions: Let $\sigma=[v_0,v_1, ... , v_p]:\Delta_p \to \Delta_q$ be an affine simplex. Let $v \in \Delta_p$ The cone on $\sigma$ from v is defined as ...
0
votes
0answers
51 views

Prove on $S^1$ deg(f)=deg(g)=>f is homoptopic to g

Let $a(t)=e^{2\pi it}$ be the generator of $\pi_1(S^1,1)$ and we define degree of $f$ this way: $[\omega^{-1}]\cdot f_*([a])\cdot [\omega]=\deg(f)[a]$ where $\omega$ is any path from $f(a)$ to $1$. ...
1
vote
1answer
76 views

Genus of a surface

Let $S$ be a genus 3 closed orientable surface. Let $R\rightarrow S$ be a degree 2 covering map. What is the genus of $S$ ? Do I have to use the Euler characteristic of a surface presentation which is ...
0
votes
1answer
20 views

Finding generators for a non-normal group

How would I find the generators for a non-normal index 3-subgroup of the free group $\langle a,b| - \rangle$ ?. I know that a finitely generated free group can be realised as the fundamental group of ...
1
vote
2answers
110 views

Covering map of a Torus

How would I draw (describe) a covering map given by $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ $T^{2}\#T^{2}\#T^{2}\#T^{2}\rightarrow T^{2}\#T^{2}$ and what would be the ...
1
vote
0answers
147 views

Lifting correspondence in Algebraic topology

Recently I have been studying algebraic topology and came across the notion of lifting correspondence. Here, lifting correspondence definition is the same that Munkres uses in his book. However, I ...
1
vote
0answers
48 views

On Dold fibration

The article of nLab on Dold fibration I have two questions: How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi:E \to B$ is a Hurewicz ...