Tagged Questions

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

239 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 ...
76 views

Generalization of invariance of domain

Suppose I say a topological space $X$ has the invariance of domain property iff $S \subset X$ is open as soon as $S$ is homeomorphic to an open set of $X$. We know that $\mathbb{R}^n$ has this ...
83 views

Prop 12.8 in Bott & Tu

Yo! This proposition in Bott & Tu have been haunting me for a year or so since I always have to come back to this book for references. More precisely, the second equality in Proposition 12.8 in ...
94 views

Is there an irreducible projective hypersurface such that its complement has zero Euler characteristic?

We know that, if $f=X_0X_1...X_n \in \mathbb{C}[X_0,...,X_n]$ and $Z(f)\subset \mathbb{CP}^n$, then the Euler characteristic of its complement is zero, i.e. $$\chi(\mathbb{CP}^n\setminus Z(f))=0.$$ ...
121 views

Does a group homomorphism up to homotopy induce a map between classifying spaces?

Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces. If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces Bf\...
156 views

When should one learn about $(\infty,1)$-categories?

I've been doing a lot of reading on homotopy theory. I'm very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the ...
106 views

172 views

A space that deformation retracts into the cylinder and Möbius band doesn't embed in $\Bbb R^3$.

Consider the Möbius band, and take the middle circle in it (so that it deformation retracts onto it). Glue the upper boundary of a cylinder through it. This gives a space that deformation retracts ...
178 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
131 views

automorphisms of the torus bundle over the circle

Let us consider a $\mathbb{T}^2$-bundle $\xi$ over circle $S^1$. These bundles are completely determined by their monodromy matrix. A fiber-preserving homeomorphism of $\xi$ is called automorphism of ...
260 views

131 views

179 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 ...
128 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 ...
125 views

Unicoherence of non-euclidean spaces

My question concerns the notion of unicoherence, which is a property that a topological space may or may not have. The definition (from Wikipedia) is: "A topological space $X$ is said to be ...
206 views

Normal subgroups of the fundamental group of a non-orientable surface.

Let $N^2_g$ be a non-orientable closed genus $g\geq 2$ surface. Is there a way to explicitly list the normal subgroups of $\pi_1(N^2_g)$ in terms of generators and relations? I am interested in ...
116 views

Identification of integration on smooth chains with ordinary integration

Let $M$ be a smooth oriented $n$-dimensional manifold and denote by $A \in H_n(M;\mathbb{Z})$ the fundamental class of $M$ (a generator of singular homology consistent with the orientation of $M$). ...
222 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 ...
186 views

Riemann-Hurwitz Formula Using Homology

Does someone know of any good reference for a proof of the Riemann-Hurwitz Formula (of Riemann Surfaces) that uses Spectral-Sequences and Homology ? Thanks in advance ! [ I think I know how to ...
829 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 ...
401 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 ...
162 views

Finding the universal cover of a matrix group

Consider the group $G = \left\{\begin{pmatrix} a & b\\ 0 & 1\end{pmatrix}: a \in \mathbb{C}^{\times}, b \in \mathbb{C}\right\} \subset GL(2, \mathbb{C})$. How does one find the universal cover ...
490 views

Fixed Points of a Reflection

This question is about problem 2.C.5 from Allen Hatcher's Topology. The statement of the problem is as follows: Let $M$ be a closed orientable surface embedded in $\mathbb{R}^3$ in such a way that ...
311 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 ...
881 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 ...
262 views

Cellular homology of the products with torus and Klein bottle with the circle

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 $K \times S^1$. Hatcher's homology notes ...
405 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 $M$ ...
168 views

Is it ever true that $\Omega^n(\bigvee_I X)\simeq\bigvee_I\Omega^n X$?

I'm curious about the following: Given a countable index set $I$, is it ever true that $\Omega^n(\bigvee_I X)\simeq\bigvee_I\Omega^n X$? What would we have to assume about $X$ (if possible) to make it ...
71 views

when a space is a product of eilenberg mclane spaces for $\pi_1(X)$ is not abelian

In When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$. , I discuss in my answer when an Abelian cw complex $X$ is a product of Eilenberg-Maclane spaces, and show that it ...
72 views

Barycentric subdivision proof

In the proof of barycentric subdivision in singular homology, we take the subdivision operator $b: C_q(X) \to C_q(X)$, and do some algebra to define an operator $b^\infty$, which applies $b$ "as many ...
204 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are simply connected manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. ...
30 views

Is $Y \cup_f X$ a CW complex?
Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the inclusion $A \to X$. My question is, is \$...