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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140 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 ...
6
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0answers
46 views

Proving that the given map has the path lifting property

Let $X$ be a locally compact metric space and $G$ be a discontinuous group of homeomorphisms of $X$. I need to show that the orbit map $p :$ $X \rightarrow X/G$ has the path lifting property. ...
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81 views

If $f : S^1 \to S^1$ is odd, then $f$ cannot be nullhomotopic

The problem is: Let $f:S^1 \to S^1$ be a continuous map that has the property that $f(-z) = -f(z)$ and $f(1) = 1$. Prove $f$ cannot be null-homotopic. Any ideas? The first step is to show that by ...
6
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0answers
121 views

Why universal G-bundles are contractible?

Let $G$ be a nice topological group and $E\to B$ a universal $G$-bundle. I'm interested in a proof of contractibility of $E$ using only the universal property of it. I also know that if there is a ...
6
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131 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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179 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 ...
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164 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 ...
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170 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 ...
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57 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 ...
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172 views

Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$. ...
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278 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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59 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
6
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83 views

Steenrod Algebra as automorphisms of additive group

Is there a direct way to see that the subalgebra of the mod-$p$ Steenrod algebra ${\mathcal A}_p$ generated by the reduced powers is isomorphic to the dual of the Hopf algebra ${\mathcal ...
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78 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 ...
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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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163 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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199 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 ...
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0answers
130 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 ...
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333 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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170 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 ...
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185 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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172 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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0answers
103 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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118 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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192 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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111 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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183 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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447 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 $B$ ...
6
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0answers
386 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 ...
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159 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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463 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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0answers
494 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 ...
6
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308 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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0answers
227 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 ...
6
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0answers
240 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 ...
6
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0answers
383 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 $M$ ...
6
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0answers
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 ...
5
votes
0answers
47 views

Connecting homomorphism is Bockstein operation, construction of natural long exact sequence.

Let $0 \to \pi \overset{f}{\to} \rho \overset{g}{\to} \sigma \to 0$ be an exact sequence of Abelian groups and let $C$ be a chain complex of flat Abelian groups. Write $H_*(C; \pi) = H_*(C \otimes ...
5
votes
0answers
96 views

Homotopy classes of functions from a finite CW complex

I am given the following problem: taken $X$ finite CW complex and $Y$ a space such that for every basepoint $y \in Y$ the group $\pi_i(Y,y)$ is finite $\forall i \leq \text{dim} X$ then the set $[X, ...
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0answers
73 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 ...
5
votes
0answers
69 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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0answers
141 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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0answers
69 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 ...
5
votes
0answers
90 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
votes
0answers
77 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 ...
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0answers
99 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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0answers
102 views

Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
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votes
0answers
40 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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0answers
92 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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0answers
40 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 ...