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

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Characteristic Class exercises

I tagged this as "homework" because my supervisor told me I need to be better at computing characteristic classes. The classic examples I can think of are tangent bundles and tautological line ...
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129 views

Why is $\pi_2(S^1 \vee S^2)$ not finitely generated?

This is an exercise following a discussion of fibrations. preceding that there was a discussion of cofibrations and the long exact sequence of homotopy groups of a pair. Any hints would be greatly ...
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152 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
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166 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
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126 views

Given a space what spaces can it cover?

I was thinking about my previous question and thought about going the other way around. Assume we are given a space $Y$ and $Y$ covers $X$, then how much can be said about $X$? The most trivial ...
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71 views

Are PL maps determined by PL-paths?

Let $X,Y$ be polyhedra. If $f\colon X\rightarrow Y$ is a PL map and $g\colon I\rightarrow X$ is a PL map (where $I$ denotes the interval), then $f\circ g$ is a PL map. Is the converse true? i.e. ...
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119 views

Homology of subsets of $\mathbb R^n$

Let $E \subset \mathbb R^n$. Must the homology groups $H_k (E)$ be trivial for $k \geq n$? How about just for $k > n$? If not, whats an example? Thanks.
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93 views

Why does applying $-\Box_{A//B} A$ to a free coresolution preserve exactness?

Let $A$ be a Hopf algebra over a field $k$, and let $B$ be a normal subHopf algebra of $A$. Suppose we have an $A$-free coresolution of $k$ over the form $F_n=K_n \otimes_k A$. Kochman claims that ...
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137 views

The action of the Steenrod algebra on $H^*(BU; \mathbb{Z}_p)$

By considering the classifying map $f \colon (\mathbb{C} P^{\infty})^n \rightarrow BU(n)$, its induced map on cohomology, and using the Cartan formula, we can derive the Wu formula for the action of ...
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153 views

Lie group quotient bundle with image of normalizer as structure group

In Glen Bredon "Topology and Geometry", Ch. II-13, I am stuck on the following "Problem 1. Finish the proof at the end of this section that $G\rightarrow G/H$ is a bundle. Also show that this is a ...
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376 views

Homology groups of unit square with parts removed

I did exercise 19 in Hatcher on page 132 and I was wondering if anyone could tell me if this is right: 19. Compute the homology groups of the subspace of $I \times I$ consisting of the four boundary ...
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760 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 ...
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301 views

Prove $e^{2 \pi k s}$ is not homotopic to constant loop at $1$ in $S^1$

Let $e^{2 \pi k s} = f(s)$, $f \colon [0,1] \to S^1$ subset of Complex numbers ($S^1$ = unit circle at origin). So $f$ is a loop at the basepoint $1$ in $S^1$. Show that it is not homotopic to the ...
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333 views

Exact sequences in the category of chain complexes

Here is the question from Rotman, verbatim: A sequence $S'_*\stackrel{f}{\to} S_* \stackrel{g}{\to} S''_*$ is exact in Comp if and only if $S'_{n}*\stackrel{f_n}{\to} S_n ...
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55 views

Computational Topology Codes

I am working on a project with a PI that thinks could be solved with computational topology tools. For this project, we will be looking at the persistent homology of objects in 3D images. I tried ...
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21 views

Properties of spectra

I've been trying to understand spectra better, and I was wondering what would be a good source to learn from. My issue with the classical treatments (e.g., Adams, Switzer) is that people keep telling ...
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34 views

Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N\to M$ between closed oriented connected manifolds. Let $X$ and $Y$ be diffeomorphic submanifolds of $M$, and assume $h$ to be ...
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65 views

Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy ...
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31 views

Homology of connected sum of CW-complexes

Let $X$ and $Y$ be finite and connected CW-complexes of dimension $n$ with exactly one $n$-cell. Then we can define their connected sum $X\#Y$ just like in the manifold case: extract an ...
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53 views

Vector Bundles over Spheres

I would like to understand how to construct a vector bundle over the n sphere give a map of its equatorial $(n-1)$-sphere into the general linear group $GL_n(\mathbb{R})$. My thought is that one ...
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66 views

$R^2\setminus K, K$ compact, is not simply connected

So, I think I'm aware of the general idea of how to do this. $K$ is compact, hence closed and bounded, so there is some circle of finite radius, say $r$, that wraps around $K$. This loop isn't null ...
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46 views

Relations between the homotopy class and the orientation of the connected sum of two manifolds

I want to find some general arguments about why it can happen that $$ M \sharp \overline{M} \not\simeq M \sharp M$$ for $M$ a compact orientable odd dimensional manifold, which is chiral, i.e. ($M ...
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29 views

Transfer homomorphism in transformation groups

I am aware of the existence of a transfer homomorphism in the setting of so called "regular $G$-complexes", as described e.g. in Bredon's Introduction to Compact Transformation Groups. But suppose ...
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36 views

Proof of Alexander Duality on Bredon - help with a passage

At page 353 of Bredon's Topology and Geometry, there is stated the Alexander Duality as Corollary $8.7$. I don't understand where does the upper row come from and why is it exact. I thought it was ...
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56 views

There is no smooth submersion from $S^2$ to $S^1$.

Show that there is no smooth submersion from $S^2$ to $S^1$. I know of one algebraic topology proof which I think is not the shortest one. That submersion is an open map should be a useful fact in ...
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38 views

Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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50 views

The relation between homotopy equivalence and contractible mapping cone?

In this MO thread, the OP claimed that it is obvious that homotopy equivalence implies the mapping cone contractible, whereas the converse proposition is wrong. I hate to admit that it's not obvious ...
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78 views

Criterion for homotopy equivalence in the category of pair of spaces

I am trying to prove the following statement : Let $\{ * \} \subset A \subset B \subset X$ be a chain of topological spaces (all subsets have the subspace topology). It is given that $A \to X$ ...
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40 views

Let $A$ be the annulus in $\mathbb{C}$, what is the space $A/{\sim}$ generated by identifying all points on the “inner circle” with each other?

The annulus is $A=\{z\in\mathbb{C}:1\leq |z| \leq 2\}$ and the "inner circle" here is the set of points $\{z\in\mathbb{C}:|z|=1\}$. If we identify all points of the inner circle together, then, ...
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118 views

Curvature of a principal bundle and the exterior covariant derivative

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...
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23 views

Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G ...
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69 views

Hatcher and “house with two rooms”

On page 4 of Hatcher's "Algebraic Topology" he constructs the "house with two rooms" space. He claims that there is some neighborhood containing this space that is homeomorphic to the unit ball (in ...
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94 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
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38 views

Graph theory application of homology

I am struggling with the idea of local homology groups and would like to see an example of how to go about finding them in general. I'm thinking of the most trivial case to apply the theory of local ...
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71 views

The set of homotopy classes of maps is in bijection with homomorphisms between homotopy groups

My question comes from the fact that $\langle K(G,n),K(H,n)\rangle = \text{Hom}(G,H)$. The only proof of this fact that I know uses $\langle X,K(H,n)\rangle = H^n(X,H)$. However, this way of ...
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30 views

Crossed product in relative cohomology

First let me fix some notations: $\Delta^p$ will be a standard $p$-simplex, $\Sigma_p(X)$ the set of all continuous maps $\sigma:\Delta^p \to X$ (where $X$ is some topological space) Let $S_p(X,R)$ ...
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101 views

The tangent bundle of $\mathbb{CP}^1$ is not isomorphic to its dual.

This question is related to this one but is not a duplicate. In " Characteristic Classes" by Milnor and Stasheff on pages 167-168, the authors give a brief argument about why the tangent bundle of the ...
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61 views

Calculating the first homology group

Suppose all vertices on a polygon are identified and the polygon is $abcb^{-1}a^{-1}c$. Is it enough to simply switch to additive notation, get $2c$ and realize that $H_1(X) = \mathbb{Z}_2 * ...
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65 views

Fundamental groups of the projective plane

I'm practicing for my topology final. Munkres' Topology, Theorem 74.4 states that: Let $X$ denote the $m$-fold projective plane, that is the space obtained by taking the connected sum of the ...
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98 views

Punctured plane

What does one point compactification of singly, doubly, triply punctured plane $\mathbb{R}^2$ look like? What would their fundamental groups look like? I'm trying to visualize but can't seem to draw ...
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47 views

cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
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143 views

Fundamental group of connected sum for non-orientable manifolds

For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$. As far as I understand, for non-orientable ...
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36 views

The images of two non homotopic to identity maps intersect

How could one prove that images two maps $f,g:\mathbb RP^4 \to \mathbb RP^7$ which are not homotopic to trivial map have nonempty intersection.
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61 views

cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...
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79 views

Lifting a sphere-valued homotopy.

Let $A\subseteq X$ be two finite cell complexes, $\dim X\leq 2n-3$ and let $[(X,A), (S^n, *)]$ be the relative cohomotopy group. There is a natural map $$ \delta: [(X,A),(B^n,S^{n-1})] \to [(X,A), ...
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64 views

Describing generators for the fundamental group of an elliptic curve given by an equation

Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a ...
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51 views

When is $H_i(X,Y)\cong H_i(X/Y)$?

For orientable manifolds,for what $X$ and $Y\not=\varnothing$ does this isomorphism hold true?
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30 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
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98 views

Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...
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50 views

quaternion vector bundle and quaternion grassmannian

Let $\mathbb{H}$ be quaternion numbers. Let $G_n(\mathbb{H}^\infty)$ be the grassmannian of $n$-subspaces of $\mathbb{H}^\infty$. Then ...