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

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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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153 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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195 views

Homology of Compact Manifolds

Perhaps this is obvious and I am overlooking something, but why are the homology groups of compact manifolds finitely generated?
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59 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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90 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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124 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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92 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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140 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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138 views

How to visualize Homology groups?

I've been studying Algebraic Topology recently (Hatcher). I have always been very good at Topology, since I am chiefly a visual type of person. I've found that most things in Topology can be thought ...
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147 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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126 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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191 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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64 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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130 views

Symmetric product of genus 2-surface

Let $\Sigma$ be the genus 2-surface. Denote $\operatorname{Sym}^2(\Sigma)=\Sigma\times \Sigma/\mathbb{Z}/2$, where $\mathbb{Z}/2$ acts on $\Sigma\times \Sigma$ by $(x,y)\mapsto (y,x)$. In the very ...
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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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119 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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252 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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71 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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67 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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181 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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204 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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94 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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70 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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489 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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187 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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130 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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89 views

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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114 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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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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232 views

How much is cohomotopy dual to homotopy?

To what degree can we dualize theorems regarding homotopy into theorems about cohomotopy (or is there a good source that tries to do this)? For instance, is there some kind of Hurewicz theorem ...
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224 views

Steenrod Algebra - Converting between Milnor to Serre-Cartan basis'

I have been studying the mod 2 Steenrod Algebra (denoted $\mathcal{A}$), using Mosher & Tangora. We have the Serre-Cartan (or Adem basis): Let $I = \{i_1,i_2,\ldots,i_n\}$ be a sequence of ...
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127 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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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 ...
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303 views

Homology groups of unit square with parts removed

I did exercise 19 in Hatcher on page 132 and I was wondering if anyone could tell me if this is right: 19. Compute the homology groups of the subspace of $I \times I$ consisting of the four boundary ...
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620 views

How to calculate all the subgroups of the fundamental group of the Klein bottle?

A problem asks me to find all the covering spaces of a Klein bottle. This needs to calculate all the subgroups of the fundamental group of the Klein bottle. But I don't have any idea how to do it. I ...
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184 views

$T \times S^1$ and $K \times S^1$ (a question from Hatcher)

Note that in what follows $T$ is the torus and $K$ is the Klein bottle I am just looking at Hatcher's example calculation using cellular homology of $T \times S^1$ and $K \times S^1$. is the image ...
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300 views

Chain map inducing isomorphism in homology

If $X$ is a CW complex, show that there is a chain map $W_*(X) \to S_*(X)$ inducing isomorphisms in homology. Here $W_p(X) = H_p(X^p,X^{p-1})$ Let $E$ be the CW decomposition of $X$ and let $M$ ...
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44 views

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
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49 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
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41 views

When is a graded ring the cohomology ring of a CW-complex?

Let $A^*$ be a graded-commutative ring with $A^n = 0$ for sufficiently large $n$ and each $A^n$ finitely generated. When does there exist a finite CW-complex $X$ with $H^*(X) \cong A$ as graded rings? ...
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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 $$ ...
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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 ...
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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 ...
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36 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
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68 views

Winding number and homotopy

Given two maps $f,g : S^1 \rightarrow S^1$, I want to show that if they have the same winding number, then there is a homotopy between them. Well, we know that we can write them as $f(\exp(2 \pi i ...
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26 views

Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
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25 views

third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$

It is known that the third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$. Each colouring $C$ of a knot diagram by a dihederal quandle $R_3$ induces a homomorphism $f:C \rightarrow ...
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44 views

The vanishing (?) cohomology of the Milnor fiber

Setup. Say we have a germ of a holomorphic function $f:(\mathbb C^{n+1},0)\to (\mathbb C,0)$ with a critical point at the origin. There is an $\epsilon>0$ small enough so that $f$ becomes a ...