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

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3
votes
2answers
71 views

every continuous map $S^1 \rightarrow S^1$ can be extended to continuous map $B^2 \rightarrow B^2$

Let $S^1$ denote the unit circle, and $B^2$ denote the closed unit disk. I came across this question and got stuck: Q:) Every continuous map $f :S^1 \rightarrow S^1$ can be extended to continuous map ...
1
vote
0answers
31 views

What is the homotopy involution on $S^2$ which is the delooping of the inversion involution on $\Omega S^2$?

Let $f:\Omega S^2\to \Omega S^2$ denote the inversion involution on $\Omega S^2$, namely, the map which sends a loop in $S^2$ to the loop in $S^2$ running in the reverse direction. Delooping gives a ...
0
votes
1answer
23 views

a region homeomorphic with klein bottle

prove that if we consider this shape in the picture below with the equivalency relation that : a & b are in one class if they are antipoles in inner or outer circles, then the induced quotient ...
6
votes
2answers
149 views

Can we simultaneously realize arbitrary homotopy groups and arbitrary homology groups?

Let's keep our groups finitely presented for the time being. All spaces in this post are path connected. Background: By a standard construction (e.g., on p. 365 of Hatcher), there exists a $K(\pi, ...
1
vote
1answer
27 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...
-1
votes
1answer
40 views

a quotient space homeomorphic with $\mathbb{R}\mathbb{P^2}$

prove that if we glue the closed unit disc to the circular boundary of the mobius strip we obtain a quotient space that is homeomorphic with $\mathbb{R}\mathbb{P^2}$. it is my general topology ...
2
votes
1answer
61 views

lifts of continuous map to covering space

The following problem gives me a bit of trouble: Let $p:E\to X$ be a covering map. Let $g_1,g_2$ be two lifts of the continuous map $f:Y\to X$. Show that $T:=\{y\in Y:g_1(y)=g_2(y)\}\subseteq Y$ ...
2
votes
1answer
55 views

Manifolds with non-vanishing vector field and vast homology

Let $n \ge 3$. Is there n-fold $M^n$ with both $\chi(M)=0$ and $\dim H_*(M,\mathbb{R}) \ge$ given number?
1
vote
0answers
36 views

constructing the klein bottle by gluing the sides of a triangle

can any one tell me why gluing the sides as in this picture would make a klein bottle for us? thank you
0
votes
0answers
44 views

one point compactification of upper half plane

"prove that the closed unit disc could be considered as the one point compactification of the upper half plane with the X-axis as the bounday" i believe in this statement by the imagination,but i ...
0
votes
1answer
39 views

Are deck transformations homotopic to the identity?

Suppose that $p: X \to Y$ is the universal covering of some connected and locally path connected space $Y$, and that $\phi$ is a deck transformation. Is $\phi$ homotopic to the identity on $X$? If so, ...
1
vote
0answers
29 views

Showing that a homotopy fiber of a fibration is homotopy equivalent to the fiber of the base point.

Assume $(E, e_0)$ and $(B, b_0)$ are based spaces with the indicated base points. Given a based fibration $p: E \rightarrow B$. We have the respective homotopy: fiber \begin{equation} Fp= ...
0
votes
1answer
33 views

A question regarding the set of one-dimensional subspaces being the same as a circle.

My Topology book says that in $\Bbb{R^2}$, the set of all one-dimensional subspaces (or lines passing through the origin) is a circle. This stackexchange question says that this is because every ...
2
votes
1answer
53 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
4
votes
1answer
39 views

Characteristic classes for quaternionic bundles

In a nutshell, the derivation of the Stiefel-Whitney and Chern classes for a (let's say) closed space $X$ is as follows: For $k = {\mathbb{R}}$ or $\mathbb{C}$, any $k$-line bundle $\xi \to X$ is the ...
1
vote
1answer
35 views

Is $\textrm{im}(f)$ homeomorphic to the torus less the inner equator?

Consider the map $id_{S^1}\times f:S^1\times [0, 1]\longrightarrow S^1\times S^1$ where $f:[0, 1]\longrightarrow S^1$ is given by $$f(t)=(\cos(\pi t), \sin(\pi t)).$$ Is it true that $\textrm{im}(f)$ ...
0
votes
2answers
54 views

A question about the rotation number of homeomorphisms of the circle

Let $f: S^1 \rightarrow S^1$ be an orientation-preserving homeomorphism of the circle and let $F: \mathbb{R} \rightarrow \mathbb{R}$ be any lift of $f$. Usually one defines the rotation number ...
2
votes
2answers
100 views

$f:X\rightarrow S^1$ a continuous map. $X$ a path-connected topological space.

Let $f:X\rightarrow S^1$ be a continuous map from a path-connected topological space $X$ and let $p:\mathbb{R} \rightarrow S^1$ be the universal covering. What is the condition when $f$ admits a ...
1
vote
0answers
31 views

What is the mapping class group of the wedge of circles?

I was wondering if there is a description of the mapping class group of a wedge of $n$ circles. Are the only kinds of homeomorphism classes in the mapping glass group are compositions of ...
3
votes
0answers
52 views

De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
1
vote
0answers
18 views

Comparison between Eilenberg-Steenrod excision and Brown representability excisive

One of the Eilenberg-Steenrod axioms for unreduced cohomology is excision, which states that $H^n(X,A)\cong H^n(X\setminus U,A\setminus U)$, for good subspaces such as when $\overset{\circ}U\subseteq ...
2
votes
0answers
35 views

Homology of smooth loop spaces of spheres

I'm looking for a reference on the homology of $C^\infty(S^1,S^n)$. So far I've only been able to find references for spaces of maps which are $C^k$ or in a Sobolev class. I also have references which ...
5
votes
2answers
86 views

Explicit expression for homeomorphism and homotopy equivalence

In many cases of topology when one needs to show two spaces $X$ and $Y$ are homeomorphic or homotopy equivalent, one uses some description instead of constructing an explicit homeomorphism or homotopy ...
3
votes
0answers
32 views

Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
1
vote
2answers
63 views

Homology of product of topological space and sphere is direct sum of homologies.

Show that for $i > n \in\mathbb{N}$: $$H_{i}\left(X \times \mathbb{S}^{n}\right) \simeq H_{i}\left(X\right) \oplus H_{i - n}\left(X\right).$$ My first idea motivated by $n=0$ case (which is ...
2
votes
1answer
60 views

Obstruction to reduction of structure group

In the wiki article it's stated that the obstruction to reduction of structure group along a morphsim $H \to G$ can be stated in terms of classifying spaces via the cofibre $BG/BH$ as follows. A ...
2
votes
1answer
31 views

Dimension of restriction of surjective linear map

I'm trying to understand the proof of theorem 4.23 (case 1) in Allen Hatcher's "Algebraic Topology". We have a map f, for which $f^{-1} : (\Delta ^{n+1})$ is a finite union of convex polyhedra, on ...
4
votes
2answers
77 views

(Certain) colimit and product in category of topological spaces

Consider the diagram $$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots $$ in category of topological spaces. Denote $I$ the ...
0
votes
0answers
24 views

the connected sum of closed orientable manifolds is orientable [duplicate]

How could I prove the following fact with singular homology theory? The connected sum of closed orientable manifolds is orientable. Thank you for your help!
1
vote
0answers
32 views

Constructing a map sending a generator of $H_n(M)$ to given generators of $H_n(M, M-U_i)$.

Let $M$ be an orientable closed manifold of dimension $n$ covered by coordinate discs $ \{ U_i : 1 \le i \le k\} $ such that for each $i$, $\bar{U_i}-U_i$ is homeomorphic to $S^{n-1}$, and suppose ...
7
votes
0answers
74 views

Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...
0
votes
1answer
38 views

About the Degree of a Map

I am reading Elements of Homotopy Theory by George W. Whitehead. In the section about maps of the $n$-sphere into itself, in the second last paragraph of the text quoted below, he says that "Then an ...
1
vote
1answer
30 views

Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
1
vote
1answer
71 views

Period of a particular finite group

Let $G$ be a group fitting in the following exact sequence: $0 \to \mathbb{Z}/p \to G \to \mathbb{Z}/q^r \to 0.$ Here $q$ and $p$ are primes (not necessarily distinct). It is easy to check (by the ...
3
votes
2answers
36 views

Generator of singular homology of n-sphere

I am learning singular homology theory right now. The homology of n-sphere is computed by Mayer-Vietoris argument. Intuitively, for example the class represented by a loop is the generator of ...
6
votes
1answer
93 views

Airy differential equation and Galois group

Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over ...
1
vote
1answer
32 views

Universal Cover of wedge sums of spaces?

I was wondering if there is some general prescription for picturing the universal cover of $X \vee Y$ for nice enough $X$ and $Y$. (Say, path-connected, locally path-connected and semilocally simply ...
1
vote
1answer
30 views

How to show $q:H_2(S^1\times S^1)\longrightarrow H_2(S^1\times S^1, S^1\vee S^1)$ is an isomorphism?

I want to solve the following exercises in Hatcher's algebraic topology book. Exercise: Show that the quotient map $S^1\times S^1\longrightarrow S^2$ collapsing $S^1\vee S^1$ to a point is not ...
0
votes
0answers
31 views

Second Volume of Elements of Homotopy Theory?

In the preface of Elements of Homotopy Theory (GTM 61) by George W. Whitehead, he wrote that "I plan to devote a second volume to these developments". Does any one know if George eventually published ...
4
votes
2answers
70 views

What is the calculus-theoretic formula to calculate the homotopy class/degree of a map $T^2\to S^2$?

I know by Hopf classification theorem that $[T^2;S^2]$(torus to sphere) are classified by the integral cohomology group $H^2(T^2;\mathbb{Z})\approx\mathbb{Z}$. Also I understand that in general, given ...
0
votes
0answers
26 views

What is “Triangulable Triad”?

I am reading George W. Whitehead's Homotopy Theory; Corollary 1.0.2 mentioned the term "Triangulable triad" without definition. May I know how it is defined?
7
votes
3answers
221 views

Symmetry of Grassmanians

I thought this might be simple (now I'm not sure) but can't solve it: why is it true that for $X,Y$ two linear $n$-subspaces of $\mathbb{R}^{n+k}$ there exists an orthogonal transformation of ...
0
votes
0answers
53 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
1
vote
0answers
33 views

A commutative diagram of (co)homology module with relative cap products

Fix a class $a\in H_{p+q}(X,A)$. Then I want to show that the diagram $$ H^p(X,A) \rightarrow H^p(X)$$ $$ \downarrow a \cap\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow a \cap$$ $$ H_q(X) \rightarrow H_q ...
0
votes
0answers
26 views

every path in $X$ is homotopic with endpoints fixed to a path passing through $b$

$X$ is path connected and b$\in$X, show every path in $X$ is homotopic with endpoints fixed to a path passing through $b$ This is the hint in the book: Let $\gamma$ be a path from $x$ to $y$. If ...
3
votes
0answers
20 views

Has there been work on computational group theory applications to computing colimits of crosses n-cubes of groups?

I'm trying to compute homotopy groups of a few spaces using crossed n-cubes of groups. I'm able to describe a few colimits in terms of quotients of induced crossed modules and nonabelian tensor ...
2
votes
1answer
17 views

$H_q(X;\mathbb{Z})=0$ when X spherical complex with $H_q(X;F)=0$ for all $q>0$ and for all $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$

Suppose that X is a spherical complex with $H_q(X;F)=0$ for all $q>0$ where $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$, as $p$ runs over all primes. I want to prove $H_q(X;\mathbb{Z})=0$. I know ...
1
vote
0answers
28 views

chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$ C^n(Y_i) \oplus C^{n-1}(X_i) $$ with boundary operator: $$ (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
2
votes
1answer
80 views

what is th homology group of $\mathbb{Q}$?

what is the 0'th homology group of $\mathbb{Q}$ I mean $ H_{0}(\mathbb{Q})$?as the 0'th homology group is counting the path component of the space so it should be infinite direct sum of copies of ...
1
vote
1answer
45 views

the fundamental group of punctured surface

Let $S_{g,m}$ be a surface of genus $g$ with $m$ punctured, we know the fundamental group of $S_{g,0}$ is $$ \pi_1(S_{g,0}) = \left\langle a_1, b_1, \dots, a_g, b_g {~\large\mid~} [a_1, b_1] \dots ...