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)

5
votes
0answers
44 views

How to kill homotopy groups using framed cobordism

Let $M$ be an orientable manifold (with or without boundary), $N$ a framed submanifold in the interior of $M$ and assume (if necessary) that $\dim N<(\dim M)/2$. If some low-dimensional homotopy ...
5
votes
0answers
62 views

There does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree.

The following is a problem from a topology qualifying exam I am studying for: Show there does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree. I think I am doing something wrong, ...
5
votes
0answers
101 views

Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
5
votes
0answers
195 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
5
votes
0answers
59 views

Homotopy type vs. weak homotopy type, and repercussions for EG

This is another basic question. I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point ...
5
votes
0answers
77 views

Serre Fibration long exact sequence

I want to know if there exists something like the long exact sequence on the following case: Let $p : E\rightarrow B$ a continuous surjective map such that there exists an open dense subset $U$ of ...
5
votes
0answers
101 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
5
votes
0answers
118 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
5
votes
0answers
84 views

Why is this space aspherical?

Let $X = Y \cup Z$ be a connected, path-connected Hausdorff space. Suppose that $Y$, $Z$, and $Y\cap Z$ are all connected, path-connected, and aspherical, and that the homomorphism $\pi_1(Y\cap Z) ...
5
votes
0answers
76 views

Leray spectral sequence for complexes

Let $f:X\rightarrow S$ be a morphism of schemes. Let $0\rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$ be an exact sequence of Abelian sheaves on $X$. Is there a general procedure to ...
5
votes
0answers
100 views

Injective Resolutions in $\mathfrak{Ab}(X)$

Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, ...
5
votes
0answers
157 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 ...
5
votes
0answers
106 views

Artin Algebraic Topology

I am reading algebraic topology by Artin, and he gives the following definition. Let $X$ be a topological space. We define a homomorphism $sd_X: S_p(X) \to S_p(X)$ by induction. If $T: \Delta_0 \to ...
5
votes
0answers
105 views

Homological definition of orientation at a boundary point?

For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
5
votes
0answers
222 views

Covering Spaces for $S^1 \vee S^1$

I'm trying to list the connected 3-sheeted covering spaces of $S^1 \vee S^1$ up to isomorphism. I know what the answer is; I've seen listings. My question is, if I'm deriving this, how do I know ...
5
votes
0answers
143 views

Inverse functor in proof of Dold Kan Correspondence

I´m looking at the proof of the Dold-Kan correspondence. Let $SA$ be the category of simplicial objects in an abelian category $A$ and $CH_{\ge0}(A) $ the category of non-negative chain complexes in ...
5
votes
0answers
130 views

Gluing manifolds

Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this: A point $(\cos \phi , \sin \phi, ...
5
votes
0answers
234 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 ...
5
votes
0answers
120 views

$\operatorname{Spin}^c(n)$ is a Lie group?

Assume that $n\geq 3$. (This implies $\pi_1(\operatorname{SO}(n))=\mathbb{Z}/2$.) Let $\rho\colon \operatorname{Spin}(n)\to \operatorname{SO}(n)$ be the 2-fold universal cover. Identify $\ker(\rho)$ ...
5
votes
0answers
138 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 ...
5
votes
0answers
193 views

Examples of special sphere bundles

I'm interested in examples of sphere bundles which do not arise from vector bundles. I'm not quite clear about the following. So please let me know if anything is false. I believe that a ...
5
votes
0answers
83 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
5
votes
0answers
127 views

diagonal image of a primitive homology class

Let $X$ be a topological space. A class $a\in H_n(X;\mathbb Z)$ is said to be primitive if $a\not = m b$ for every integer $m>1$ and $b \in H_n(X;\mathbb Z)$. Let $$\Delta_*:H_n(X;\mathbb Z)\to ...
5
votes
0answers
260 views

Cohomology of fiber bundle with a section

Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map $$ ...
5
votes
0answers
187 views

Homotopy equivalence in the category of arrows.

I'm reading Jeff Strom's book on Homotopy Theory and I am trying to make some sense of a certain exercise. On page 91, "Homotopy in Mapping Categories" we consider the category of arrows of ...
5
votes
0answers
303 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
5
votes
0answers
188 views

A counter example in obstruction theory

Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map ...
5
votes
0answers
105 views

Fundamental group of the complement of a linear subspace

Let $m<n-1$ be two positive integers. Consider $\mathbb{R}^m$ as a subspace of $\mathbb{R}^n$ via $\mathbb{R}^m\times \{(0,0,...0)\}$. Any suggestions on how to compute ...
5
votes
0answers
295 views

Definition of Cobordism

I am currently trying to read Quillen's Cobordism paper - On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298. Still on the first page ...
5
votes
0answers
245 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 ...
5
votes
0answers
290 views

homology groups, physical interpretation

One can roughly interpret Betti numbers as the number of n-dimensional holes of our space. Is there a reasonable interpretation for torsion coefficients? Moreover, for 'physical' applications, are ...
5
votes
0answers
183 views

Ring structure in the Serre spectral sequence

I've tried to understand what's going on in Example 1.5 on page 27-28 in Hatcher's notes on spectral sequences. There is one part in the reasoning that I can't understand here. He writes down a table ...
5
votes
0answers
151 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
votes
0answers
316 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
votes
0answers
32 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
4
votes
0answers
36 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 ...
4
votes
0answers
103 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
votes
0answers
137 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 ...
4
votes
0answers
141 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 $x_1$ ...
4
votes
0answers
42 views

Compute the induced map on $\mathbb{CP}^n$

Let $d>0$ and $f:\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$ be given by $f(z_0,...,z_n)=(z_0^d,...,z_n^d)$. Let $F:\mathbb{CP}^n \rightarrow \mathbb{CP}^n$ be the induced map by $f$. Compute ...
4
votes
0answers
71 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
votes
0answers
84 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 ...
4
votes
0answers
57 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
votes
0answers
70 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 ...
4
votes
0answers
76 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
votes
0answers
54 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
votes
0answers
54 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 ...
4
votes
0answers
31 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
votes
0answers
52 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 ...
4
votes
0answers
44 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 ...