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

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2
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2answers
97 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
51 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
17 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
81 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
62 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
58 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
76 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
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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
35 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
91 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
30 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
69 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
25 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
215 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
52 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
31 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
25 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
77 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
43 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 ...
2
votes
1answer
25 views

Degree of a restriction of a continuous map?

I have a map $f:D^2 \rightarrow S^2$ and $f(-x)=-f(x)$ for $x \in S^1$. Does this mean that $\deg(f|_{S^1})=0$? if so, why? We defined this degree on $S^1$ as $f(\exp(t))=\exp(F(t))$ then ...
1
vote
2answers
70 views

Fundamental groups and some properties

I have some basic questions about fundamental groups that came up when I tried to prove a few things: I am sorry that they are kind of informal questions, but I could not find any answers to them in ...
4
votes
0answers
70 views

Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
0
votes
0answers
12 views

Acyclic Model Theorem Application

I want to apply $acyclic$ $model$ $theorem$ and need to check some properties. Define $C\times C$ be the category who's objects are ordered pairs $(X,Y)$ and morphisms are ordered pairs $(f,f')$, ...
-1
votes
1answer
33 views

Associative property of free product of groups

I am reading Algebraic Topology by Allen Hatcher (available online at http://www.math.cornell.edu/~hatcher/AT/AT.pdf) and at line 1 of page 42, it reads: "... because of the relation ...
0
votes
0answers
42 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
-1
votes
1answer
55 views

Does there exist a “Möbius cloth”?

If such a cloth exists, then one should no more worry about the orientation of our clothes, which troubled me sometimes. :P Thus I am wondering Does there exist a non-orientable surface with $3$ ...
0
votes
2answers
52 views

Fundamental group of torus by van Kampens theorem

So I am currently going through some lecture notes where the fundamental group of a torus is calculated by van Kampen's theorem: The torus is decomposed into its characteristic fundamental polygon ...
0
votes
1answer
45 views

Questions about van Kampen's theorem.

I just read some things about van Kampen's theorem that threat this one from a different perspective than we discussed in class and this brought up a few questions: It was said that the images of the ...
3
votes
1answer
45 views

How do you construct the lifted topology of a groupoid cover?

If I have a particularly nice space $X$ (Hausdorf, locally path connected, semi-locally 1-connected, I think), then then there is an equivalence of categories between the cover category over $X$ and ...
0
votes
1answer
51 views

Determine if these spaces are connected, Hausdorff, or compact.

Let $X = [0,1]/(0,1)$ and let $\pi: [0,1] \rightarrow X$ be the quotient map. Answer the following questions, proving your assertions: a) Is $X$ contractible? We need $s:X \rightarrow ...
0
votes
0answers
18 views

Understanding definition of properly discontinuous action

From Bredon, we say that the $G$-action on a space $X$ is properly discontinuous if "Each point $x \in X$ has a neighborhood $U$ such that $g(U) \cap U \neq \emptyset$" implies "$g = e$, an ...
3
votes
1answer
77 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
1
vote
0answers
68 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
1
vote
2answers
57 views

boundary of $M \times I$ where $M$ is the Möbius band

Let $M$ be the Möbius band and $I$ be the closed interval $[0,1]$. What is the boundary of $M \times I$? Is it orientable? What can I do when I want to know the boundary of such space? Please give an ...
0
votes
2answers
52 views

How to compute Euler characteristic from polygonal presentation?

How can I compute the Euler characteristic of a compact surface from its polygonal presentation $\langle S | W_1 , \ldots , W_k \rangle$? I guess that the number of edges is the number of different ...
2
votes
0answers
26 views

A correspondence between generators of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ and eq. classes of orthonormal frames

The problem is about (topological) orientation of $\mathbb{R}^n$: Define an equivalence relation on orthonormal frames in $\mathbb{R}^n$ by declaring two frames equivalent if the matrix expressing ...