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

learn more… | top users | synonyms (1)

4
votes
0answers
185 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?
4
votes
0answers
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 ...
4
votes
0answers
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 $\pi_i(S^j)$ ...
4
votes
0answers
88 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 ...
4
votes
0answers
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 ...
4
votes
0answers
89 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) = ...
4
votes
0answers
139 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. ...
4
votes
0answers
136 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 ...
4
votes
0answers
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, $g_k \in \mathbb{Z}$, ...
4
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
63 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 ...
4
votes
0answers
210 views

Grothendieck on (topological) Chern Classes

I have been reading through the wikipedia article about Chern classes and it currently has a section devoted to the Alexander Grothendieck axiomatic approach. The language used throughout the section ...
4
votes
0answers
128 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 ...
4
votes
0answers
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 ...
4
votes
0answers
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 $S^1$ ...
4
votes
0answers
251 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 ...
4
votes
0answers
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 $t_0≤t≤1$. ...
4
votes
0answers
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 ...
4
votes
0answers
180 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 $H_1(F,\mathbb R)$, ...
4
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
483 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 ...
4
votes
0answers
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 ...
4
votes
0answers
186 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 ...
4
votes
0answers
128 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 ...
4
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
229 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 ...
4
votes
0answers
220 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 ...
4
votes
0answers
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 ...
4
votes
0answers
144 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 ...
4
votes
0answers
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 ...
4
votes
0answers
617 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 ...
4
votes
0answers
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 ...
4
votes
0answers
297 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 $M$ ...
3
votes
0answers
48 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 ...
3
votes
0answers
39 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? ...
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 $$ ...
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 ...
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
38 views

what is the homology groups some quotient space of torus

what is the homology group for The quotient space of $S^1 \times S^1$ obtained by identifying points in the circle $S1 \times\{x_0\} $ that differ by $\frac{2 \pi}{m}$ rotation and identifying points ...
3
votes
0answers
25 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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
35 views

Cohomology of $S^2\times S^2/\mathbb{Z}_2$

The product of two spheres admits a diagonal $\mathbb{Z}_2$-action, $(x,y)\mapsto (-x,-y)$. I'm trying to compute the integral singular cohomology ring of the orbit space $X$ of this action. $X$ is ...