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

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5
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158 views

Invariance of Wall's self-intersection under the regular homotopy

For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
5
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345 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 $M$ ...
4
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25 views

How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation $$B_n = \langle \sigma_1, \cdots, \sigma_{n-1} | \ \ \sigma_i\sigma_j = \sigma_j\sigma_i\ \ (|i - ...
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55 views

Proof of the isomorphism $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2).$

I want to prove the following: $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2)$ via the map $[f] \mapsto f^* w_1(\gamma^1)$ where $\gamma^1$ denotes the tautological line bundle over ...
4
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86 views

Yoneda-type lemma for compositions on the hom-functor

The Yoneda lemma basically rephrases a rigidity property of natural transformations out of a covariant hom functor: A natural transformation $\psi : \mathsf{Hom}(Z,-)\Rightarrow F$ is determined by ...
4
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80 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
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15 views

Covering space of surface of infinite genus

Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free ...
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42 views

Inductively Constructing Chain Homotopies

Let $f,g:C_\bullet \rightarrow D_\bullet$ be chain maps. Let $T$ be a generator (possibly not the only one) of $C_n$ and assume $H_n(C)=0$. I'm trying to prove that for every $T$ there's a solution ...
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62 views

a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

I've been wondering about such problems. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$. But does there exist a space $X\subset\mathbb R^3$ (maybe even ...
4
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76 views

Hermitian Matrices with At Most Pair-wise Eigenvalue Degeneracy

Let $n\in2\mathbb{N}$ be given. Let $H\in Mat_{n\times n}(\mathbb{C})$ be a Hermitian traceless matrix such that its eigenvalues have at most pairwise degeneracy. (That is, if the eigenvalues are ...
4
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55 views

representatives of $\pi_n(U(n))$

I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps ...
4
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144 views

On the path-connectedness of $X_{i} \setminus X_{i-2}$

Suppose $X$ is an $n$-dimensional regular CW-space (a space with a regular CW decomposition). What are the weakest sufficient conditions required on $X$ to ensure that all regular CW decompositions ...
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39 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
4
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57 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
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51 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
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207 views

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
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47 views

Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit ...
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117 views

De Rham Cohomology is Sheaf Cohomology

I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold. I would like to do this without using Čech ...
4
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153 views

Standard spaces with non-standard topology and around

The following series of questions gives me no rest. Let $\mathbb{\widetilde{R}}^n$ be $\mathbb{R}^n$ with Zariski topology, i.e. we say that $A\subset \mathbb{R}^n$ is closed if it is given by the ...
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43 views

About the Chern class of infinite complex Grassmannian

I learned that any characteristic class of rank-$k$ complex vector bundles on paracompact spaces is determined bijectively by a cohomology class in $H^*(Gr_k^\infty(\mathbb C))$, the cohomology ring ...
4
votes
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158 views

Higher homotopy groups: Basepoint independence.

Let $f,g: [0,1]^n=I^n \rightarrow X$ be contiuous maps s.t. $f(\partial I^n)=g(\partial I^n)=x_1$. If $\gamma:I \rightarrow X$ is a path joining $x_0$ and $x_1$ ...
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179 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
4
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98 views

A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
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82 views

Holomorphically simply connected implies simply connected

In my book on complex analysis a "Holomorphically simply connected" set is defined as a set where for any holomorphic function $f $ and any closed path $\gamma _1 $ we have that $\int_{\gamma _1 } ...
4
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89 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
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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 ...
4
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58 views

Does $\pi_0$ commute with sequential colimits?

I'm looking for an example of a sequence of topological spaces $Y_1 \rightarrow Y_2 \rightarrow \cdots$ such that the induced map $$ \text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i) $$ ...
4
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86 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 ...
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54 views

Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then ...
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33 views

Isomorphism of the etale fundamental group

Given a birational proper morphism $f\colon X \rightarrow Y$ of complex algebraic varieties. It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(Y)$ is an isomorphism? I think ...
4
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62 views

Fiber Bundle of Manifolds

I want to show the following: Let $E \rightarrow B$ be a fiber bundle with fiber $F$. Show that if $B$ and $F$ are manifolds, then so is $E$. Solving this problem seems easy enough simply by ...
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52 views

Reference about history of characteristic classes

I'm looking for a good reference about history of characteristic classes. For example, I would like to know how Chern define the Chern class at first in history, or who rewrote the characteristic ...
4
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107 views

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 ...
4
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0answers
54 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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65 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 ...
4
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0answers
54 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 ...
4
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157 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 ...
4
votes
0answers
80 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 ...
4
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88 views

Products of homology groups

In the topology course I attend we work with the following, rather unusual definition of homology groups: For topological spaces $X,Y$ define the presheaf $\mathcal{F}_Y$ on $X$ as follows: Let ...
4
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0answers
86 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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0answers
67 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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74 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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64 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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0answers
102 views

deck transformations and covering spaces

Let $p:\tilde X\rightarrow X$ be a universal covering space, and let $H\leq G$ where $G$ is the group of covering transformations. Let $q:\tilde X \rightarrow \tilde X/G$ be the quotient map which is ...
4
votes
0answers
161 views

Leray-Hirsch theorem

I'am studying the book "Bott, Tu Differential forms in algebraic topology." I don't understand the proof of Leray-Hirsch theorem via Cech-de Rham complex. Lets consider some bundle $\pi: E \mapsto ...
4
votes
0answers
166 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 ...
4
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0answers
61 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 $\pi_i(S^j)$ ...
4
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0answers
142 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
106 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) = ...