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

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0
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1answer
84 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, ...
2
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0answers
65 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= ...
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1answer
40 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
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1answer
86 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
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1answer
83 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 ...
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1answer
37 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 $\textrm{im}(f)$ ...
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2answers
83 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
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2answers
115 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 ...
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0answers
38 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
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0answers
74 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 $$ ...
3
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0answers
35 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 ...
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0answers
40 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 ...
4
votes
2answers
117 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
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0answers
51 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 ...
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2answers
100 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
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1answer
96 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
40 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
99 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 ...
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0answers
27 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!
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0answers
34 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 ...
13
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0answers
182 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
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1answer
39 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 ...
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1answer
44 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
79 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
130 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
108 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
183 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
41 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 ...
1
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1answer
63 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
86 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 ...
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0answers
66 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
258 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
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0answers
88 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
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0answers
43 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 ...
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0answers
43 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
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0answers
25 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
23 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 ...
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0answers
118 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
104 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
142 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
41 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 ...
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2answers
95 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
81 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
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0answers
22 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 $(f,f')$, ...
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1answer
51 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 ...
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0answers
77 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 ...
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1answer
65 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
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2answers
134 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
58 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 ...
4
votes
1answer
77 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 ...