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

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A question about the universal coefficient theorem.

Or rather a couple of questions. Let $X$ be some topological space, $R$ be a (unital) PID and $G$ be an $R$-module. Am I correct in understanding that the singular cochain complexes ...
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42 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
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43 views

what can you say about the degree of $f:\mathbb{C}P^n \to \mathbb{C}P^n$

Any thoughts on this problem: If $M$ and $N$ are simply-connected, $n$-dimensional manifolds, then $H^n(M;\mathbb{Z}) \cong \mathbb{Z} \cong H^n(N;\mathbb{Z})$. A map $f:M \to N$ induces a map ...
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142 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
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50 views

Original proof of the Invariance of Domain Theorem (in English)?

Does anyone know where I can find a translation of the original proof of the Invariance of Domain Theorem in English? Wikipedia cites the original proof to be in: Beweis der Invarianz des ...
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138 views

is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
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146 views

Constructing $\pi_1$ actions on higher homotopy groups.

I am working on exercise 4.2.7 of Hatcher, which is to construct a CW complex $X$ with arbitrary homotopy groups and a prescribed action of the fundamental group on these homotopy groups (so making ...
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131 views

Relationship between paradoxes in logic and geometry/topology

Though I've been reading for years, this is my first question here. Believe it or not, I've tried the search feature- apologies if this is a duplicate. The main point of this post can be summarized ...
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71 views

Homotopic maps to $S^n$

I'm working through a proof that, given an oriented compact (connected) $n$-manifold $M$ with boundary, any two continuous maps $f,g:M\to S^n$ are homotopic. The proof uses the double of $M$, which is ...
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79 views

CW structure on infinite products

There is a standard CW-topology on the finite product $X\times Y$ of CW-complexes $X$ and $Y$. Is there a standard CW-topology on an infinte prodcut $\prod_{n=1}^{\infty}X_{n}$ of CW-complexes? With ...
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125 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
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61 views

Understanding J homomorphism

I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism. It's clear that an element of $\pi_r(SO(q)$ ...
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68 views

$S^{2n}$ is the universal cover of $B$, what is $\pi_1(B)$.

Some students and I have tried to solve this problem in the following ways: Using degree theory and results about deck transformations. Using that $S^{2n}$ is the covering space of ...
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141 views

Simplicial homology for n-simplex

I've just started to study homology theory. And I'm trying to calculate all $H_n(\Delta_N)$ for some $N$. I know that the number of $m$-simplex in $N$-simlex is $b_{N,m}={N+1 \choose ...
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97 views

What is $K (S^1,1)$?

It is known that if $G$ is a discrete group, then $BG= K(G,1)$. Originally I was interested in comparing classifying spaces of topological groups with the classifying spaces of the same groups ...
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63 views

Does an equivariant weak equivalence induce weak equivalences on all orbits?

This question arose from another, which was not well formulated and completely answered by this MO thread as pointed out by the user roman. Let $G$ be a discrete group, $X$ and $Y$ be $G$-spaces (and ...
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158 views

Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
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60 views

$J$-homomorphism and homotopy

We have Bott periodicity theorem for unitary group $U(n)$: $$ \pi_{i-1}^{s}(U) = \pi_{i-1}(U(m)) \simeq \pi_{i}(Gr_m(\mathbb{C}^{2m})) \simeq \pi_{i+1}(SU(2m)) \simeq \pi_{i+1}^{s}(U) .$$ So we can ...
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58 views

Which is harder to compute: $\pi_{n+k}$ or $\Omega^{fr}_n$?

Denote the $n+k$-th homotopy group of $S^n$ by $\pi_{n+k}(S^n)$ and the group of framed cobordism classes by $\Omega_n^{fr}(S^k)$. A central problem of algebraic topology is to compute $\pi_i(S^j)$ ...
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92 views

Perverse sheaves (or D-modules) on vector spaces, constructible with respect to a hyperplane arrangement

Let $V$ be a finite dimensional complex vector space and let $\mathcal{A}$ be a finite collection of hyperplanes in $V$. Stratify $V$ by the intersections of elements of $\mathcal{A}$, and consider ...
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133 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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98 views

The classifying space of a gauge group

Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by $$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
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143 views

Show that $f$ is a homeomorphism of $X$ onto $f(X)$

I am having trouble on the following question. Some help would be much appreciated. Let $X$ be a Hausdorff space and let $f: X \rightarrow \mathbb{R}^n$ be a proper injective continuous function. ...
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178 views

Show that degree of constant map is zero

Let $f \colon S^n \to S^n$ be a constant map, $n > 0$. I want to show that $\deg f = 0$. I will do it by definition. Let $\sum_k g_k \sigma^n_k$ be a singular chain, $g_k \in \mathbb{Z}$, ...
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132 views

Calculating H_0 directly from Eilenberg-Steenrod axioms

It's well-known that every homology theory satisfying Eilenberg-Steenrod axioms is isomorphic to singular homology. I tried to perform some homology calculations directly from axioms but couldn't do ...
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201 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
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67 views

How to use the Prontrjagin-Thom construction to obtain the Gysin map?

I need help to understand the diagram in Miller's script Vector Fields on Spheres, etc. Chapter 23, p.82 on the bottom of the page. Before, Miller introduces the Prontrjagin-Thom construction: It is ...
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62 views

Computing number of path components.

Let $H \subset \mathbb{R}^n$ be a closed subset homeomorphic to $\mathbb{R}^{n - 1}$. Could you give me an idea of how to prove that $\mathbb{R}^n - H$ has exactly two path components. I am expecting ...
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506 views

Fundamental group of an orientable surface of infinite genus.

I am trying to calculate the fundamental group of an orientable surface $X$ of countably infinite genus. The $1$-skeleton $Y$ of $X$ is infinite wedge of circles, so its fundamental group is free ...
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120 views

Transforming the Dirac Operator on $S^1$

My goal is to understand as much as I can about the Dirac operator on $S^1$ where we give $S^1$ the spin structure given by the connected double cover of the frame bundle. The spinor bundle on $S^1$ ...
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298 views

Fundamental group of the complement of Borromean rings

I got a homework asking me to show the fundamental group of the complement of the Borromean rings (say, $R^3/B$) I know there should be three generators, but I had a hard time to find the relations; I ...
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73 views

Contractible and Compact space can be contained in an open set after time $t_0$?

$X$ is a topological space that is contractible and compact. Show that if $U$ is an open set in $X$ containing $x_0$ then there exists $t_0<1$ so that $H(x,t)∈U$, for all $x∈X$, and all $t_0≤t≤1$. ...
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72 views

homotopy type of the closure of a subset

Let $X$ be a topological space and $N$ a subset of $X$. Is it true that the closure of $N$ in $X$ is homotopy equivalent to $N$. I think it is not. take for example $N=\mathbb Q\subset \mathbb ...
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185 views

Homology and Homotopy group

$E, F, B$ are topological space, $B$ path connected. If we have given a long exact sequence.. $$\cdots\to \pi_1(F)\to \pi_1(E)\to \pi_1(B)\to\cdots$$ what will the relationship of $H_1(F,\mathbb R)$, ...
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207 views

Morse theory and homology of an algebraic surface (example)

Let $T_n$ denote the $n$-th Chebyshev polynomial and define $f_n(x,y,z)\!:=\!T_n(x)\!+\!T_n(y)\!+\!T_n(z)$ and $$Z_n:=\mathcal{Z}(f_n) \subseteq \mathbb{R}^3,$$ the Bachoff-Chmutov surface, where in ...
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99 views

Dixmier-Douady class: computations

As far as I know, Dixmier-Douady classes represent obstrucions to spin$^c$ structures. Questions: Could somebody prove or give a reference: manifolds of dimension lower than $5$ always have a ...
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72 views

Coboundary of Thom class and Thom class of boundary

In Griffiths and Morgan's book "Rational Homotopy Theory and differential forms", pages 154-158, they give an example of a computation in de Rham cohomology of the minimal model of a DGA using Massey ...
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549 views

Torsion in homology groups of a topological space

It seems as though "nice" spaces don't have torsion in their homology groups. What is the underlying characteristic of these nice spaces; that they can be embedded in $\mathbb{R}^3$? So what are some ...
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1k views

Torus as double cover of the Klein bottle

Reading through some lecture notes and it says The torus $T^2$ is the orientation double cover of the Klein bottle $K$, via the covering projection $p:T^2\to K; [x,y]\mapsto [x,2y]$ Could someone ...
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199 views

Characteristic Class exercises

I tagged this as "homework" because my supervisor told me I need to be better at computing characteristic classes. The classic examples I can think of are tangent bundles and tautological line ...
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110 views

Why is $\pi_2(S^1 \vee S^2)$ not finitely generated?

This is an exercise following a discussion of fibrations. preceding that there was a discussion of cofibrations and the long exact sequence of homotopy groups of a pair. Any hints would be greatly ...
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137 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
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471 views

Hatcher 1.3. problem 16

Given maps $X\to Y\to Z$ such that both $Y\to Z$ and the composition $X\to Z$ are covering spaces, show that $X\to Y$ is a covering space if $Z$ is locally path-connected.
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Decomposition of vector bundles over a CW complex

Let $X$ be CW complex having only cells up to dimension $n$. I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a ...
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117 views

Given a space what spaces can it cover?

I was thinking about my previous question and thought about going the other way around. Assume we are given a space $Y$ and $Y$ covers $X$, then how much can be said about $X$? The most trivial ...
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115 views

Homology of subsets of $\mathbb R^n$

Let $E \subset \mathbb R^n$. Must the homology groups $H_k (E)$ be trivial for $k \geq n$? How about just for $k > n$? If not, whats an example? Thanks.
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93 views

Why does applying $-\Box_{A//B} A$ to a free coresolution preserve exactness?

Let $A$ be a Hopf algebra over a field $k$, and let $B$ be a normal subHopf algebra of $A$. Suppose we have an $A$-free coresolution of $k$ over the form $F_n=K_n \otimes_k A$. Kochman claims that ...
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130 views

The action of the Steenrod algebra on $H^*(BU; \mathbb{Z}_p)$

By considering the classifying map $f \colon (\mathbb{C} P^{\infty})^n \rightarrow BU(n)$, its induced map on cohomology, and using the Cartan formula, we can derive the Wu formula for the action of ...
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163 views

Properties of Surgery on Manifolds?

I am trying to give a brief explanation in which I make use of the concept of surgery on an $m$-manifold $M$. This is along the lines of (and taken generously from) the Wikipedia entry on Surgery; ...
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145 views

Lie group quotient bundle with image of normalizer as structure group

In Glen Bredon "Topology and Geometry", Ch. II-13, I am stuck on the following "Problem 1. Finish the proof at the end of this section that $G\rightarrow G/H$ is a bundle. Also show that this is a ...