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

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6
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74 views

An h-cobordism problem

Im trying to understand the proof of Lemma 2.3 of Milnor and Kervaire: Groups of homotopy spheres I. Suppose we have a simply connected manifold $M$ which bounds a contractible manifold $W'$. Then ...
6
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70 views

Automorphism on a kahlerian variety which acts as -id on $H^2(X,\mathbb{Z})$

I've read this proposition: "there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity" so I tried to prove this statement and ...
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137 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, ...
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193 views

How to classify principal bundles over a 2 dimensional surface?

I just want to how much people know about this at the moment? I thought this is elementary and may be of execrise level, but a quick google search showed serious papers written on this subject (like ...
6
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288 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 ...
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165 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 ...
6
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167 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, ...
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166 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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375 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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96 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 ...
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114 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 ...
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172 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 ...
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108 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 $M$). ...
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174 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 ...
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403 views

Mayer-Vietoris implies Excision

Assume $H_n$ is a covariant homotopy functor on the category of locally compact Hausdorff spaces which has the Mayer-Vietoris property: whenever $X$ is the union of two closed subspaces $A$ and $B$ ...
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155 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 ...
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397 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 ...
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455 views

Homotopy extension property vs. good pairs

I'm taking a course that uses the book "algebraic topology" by Allen Hatcher. I this book there are two different ways in which a pair (X,A) of a topological space X and a subspace A can be nice: They ...
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301 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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304 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 ...
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219 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 ...
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225 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 ...
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167 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 ...
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26 views

Based space, commuting in diagram up to homotopy.

Theorem. For any based space $Z$, the induced sequence$$\dots \to [Z, \Omega F f] \to [Z,\Omega X] \to [Z, \Omega Y] \to [Z, Ff] \to [Z, X] \to [Z, Y]$$is an exact sequence of pointed sets, or of ...
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55 views

Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
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137 views

Is $\mathbb{S}^1 \wedge E$ a cofinal subspectra in $\Sigma E$?

I'm following the proof of Switzer's "Algebraic Topology and Homotopy" of the known result Theorem. Let $E$ be a (CW-pre)spectra. There is a natural (up to homotopy) homotopy equivalence $E \wedge ...
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61 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
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0answers
76 views

Minimum number of sets required for a good open cover

A good open cover of a topological space is an open cover such that all open sets in the cover, and all finite intersections of open sets in the cover are contractible. For example, $S^2$ has an ...
5
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0answers
64 views

“Toys” spaces in algebraic topology

I did follow a course of algebraic topology last semester and I still want to continue to do some computations. But in many books it's all the time the same examples which comes back for computing ...
5
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0answers
78 views

How do the direct and inverse image sheaf functors interact with homotopy?

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for ...
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32 views

Topological invariants that detects change of orientation for an odd dimensional manifold.

Assume all manifolds involved to be closed and orientable. During my studies I learnt about the signature of a $4k$-manifold, and it turned out that (when it is non zero) it can be use to make a ...
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80 views

Twisted nail puzzle framed in terms of algebraic topology?

See here for a description of the puzzle: Twisted Nail Puzzle. My question is, can someone provide a description of the puzzle and its solution in context of the language of algebraic topology?
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36 views

Homeomorphism between boundaries of open sets is “surjective”

Let $U,V\subset\mathbb{R}^n$ two bounded open sets and let $F:\overline{U}\to\mathbb{R}^n$ be a continuous map. Assume that $F$ maps $\partial U$ homeomorphically onto $\partial V$. The question: is ...
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98 views

Hatcher 2.1.10…

Hatcher asked a question in chapter $2$ (a) Show the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic ...
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0answers
97 views

An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
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0answers
28 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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70 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 ...
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71 views

Is local homology group on a manifold a sheaf?

Let $X$ be a manifold of dimension $n$, and define $\mathcal{F}(U) = H_n(X,X-U)$. Then clearly $\mathcal{F}$ is a presheaf. I am thinking whether $\mathcal{F}$ is a sheaf. According to Lemma 3.27 in ...
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217 views

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in ...
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0answers
111 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
5
votes
0answers
243 views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
5
votes
0answers
47 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
5
votes
0answers
59 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 ...
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95 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 ...
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124 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
234 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 ...
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87 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
102 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
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0answers
119 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 $H^2(E,\mathbb{Z})$.?
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177 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...