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

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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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65 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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29 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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97 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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63 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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46 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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137 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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34 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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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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63 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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50 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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29 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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96 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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47 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 ...
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73 views

Few questions about compactly generated Hausdorff spaces and Yoneda lemma.

It is a well known fact that in CGWH spaces we have following homeomorphism: $$\mathrm{map}(X \times Y,Z)\cong\mathrm{map}(X, \mathrm{map}(Y,Z)).$$ The proof starts like this: We are dealing with ...
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24 views

examples of k-invariants of spectra

The homotopy groups of commonly used topological spectra (like KO, S, MO, MSO, etc) are easy to find in literature, even appearing on Wikipedia's List of Cohomology Theories; however, I have had some ...
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53 views

How do I show that these two presentations are isomorphic?

I'm taking algebraic topology this year and my professor assigned a class an exercise, and he told that the exercise will be on a coming exam. The exercise is: Show that $(x,y|xyx=yxy)\cong ...
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35 views

Holomorphic $K$-theory

$K$-theory is traditionally defined for arbitrary compact Hausdorff spaces. If instead we require the base to be a complex manifold and work with only holomorphic vector bundles, in what ways would ...
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67 views

Definition of the triad homotopy groups

Let $ \ n \in \mathbb{N}$, $n \geqslant 2$. Equip $\mathbb{R}^n$ with the usual euclidean norm $ \ \Vert . \Vert : \mathbb{R}^n \to \mathbb{R}$, metric and topology. Consider $ \ p_0 = (1,0,0,...,0,0) ...
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62 views

Homology of stunted infinite real projective space

Consider the following composite based map $$f: S^2 \xrightarrow{\sim} RP^2/RP^1 \to RP^\infty/RP^1$$ induced by the inclusion of the real projective plane $RP^2$ into infinite real projective space ...
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63 views

About homotopy fiber at Hatcher's book

What is the meaning of the statement: In this case a map $(I^{i+1},∂I^{i+1},J^i) \to (B,A,x_0)$ is the same as a map $(I^i,∂I^i) \to (F_f, \gamma_0)$ where $\gamma_0$ is the constant path at ...
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Can one prove that $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$ without invoking the long exact sequence of a fibration?

In trying to remind myself why $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$, the natural argument that comes to mind is to take the long exact sequence associated to the fibration $S^1 \rightarrow S^\infty ...
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39 views

Group bundle over a topological space

Suppose $p:\tilde X\rightarrow X$ is the universal cover of $X$. Take $G$ a group where $\pi_1(X,x)$ acts by isomorphisms. I read that if we consider $X\times G$ ($G$ with the discrete topology) and ...
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46 views

Cycles non-homologous but with the same winding number at each point outside?

Problem (Fulton's Algebraic Topology: A First Course, Problem 6.28) Can we find a closed set $X\subseteq\mathbb R^2$, such that there's a cycle $\gamma$ in $X$ not homologous to zero but for each ...
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68 views

Brief summary of simplicial, CW and manifold notions

I tried to summarize the relations between the following notions of: a manifold (smooth, topological and PL), simpilicial complex, CW complex. However I found some inconsistencies, which may be not a ...
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77 views

Fundamental group of 7-gon with labelling scheme $abaaab^{-1}a^{-1}$

I want to calculate the fundamental group of of a $7$-gon with labelling scheme $abaaab^{-1}a^{-1}$. I will call the quotient space $X$ with reference point $x_0$. This is what I tried: If you have ...
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60 views

Is there fundamental group “with coefficients”

Constructing fundamental group in usual way, one uses $I=[0,1] \in \mathbb R$, so uses real numbers, and gets a group classifying coverings for nice spaces: locally connectible and so on, in fact ...
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Homology of non-singular projective algebraic variety

I am unsure whether or not the following claim is true or false and whether or not my proof works or not: Claim: Let $V \subset \mathbb{C}P^n$ be a complex $k$-dimensional, non-singular, projective ...
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Higher homotopy groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
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88 views

Motivation behind definition of homologous cycles

Two cycles are said to be homologous if their difference is a boundary.(usual meanings implied) What is the motivation behind this definition or the intuitive meaning it carries. I am looking of ...
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41 views

$\pi_0$ of $M(2) \wedge M(2)$

My motivation is trying to understand Tom Goodwillie's argument here: http://mathoverflow.net/questions/87919/difficulties-with-the-mod-2-moore-spectrum and the only thing I don't get is why ...
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49 views

Coboundary map problem

I have to show that if $A$ and $B$ are compact connected subsets in the plane such that $A\cap B$ is not connected (and not empty), then $\Bbb R^2\setminus(A\cup B)$ is not connected. The tool I must ...
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56 views

When is a graded ring the cohomology ring of a CW-complex?

Let $A^*$ be a graded-commutative ring with $A^n = 0$ for sufficiently large $n$ and each $A^n$ finitely generated. When does there exist a finite CW-complex $X$ with $H^*(X) \cong A$ as graded rings? ...
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78 views

De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
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44 views

Comparison between Eilenberg-Steenrod excision and Brown representability excisive

One of the Eilenberg-Steenrod axioms for unreduced cohomology is excision, which states that $H^n(X,A)\cong H^n(X\setminus U,A\setminus U)$, for good subspaces such as when $\overset{\circ}U\subseteq ...
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58 views

Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
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25 views

Has there been work on computational group theory applications to computing colimits of crosses n-cubes of groups?

I'm trying to compute homotopy groups of a few spaces using crossed n-cubes of groups. I'm able to describe a few colimits in terms of quotients of induced crossed modules and nonabelian tensor ...
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64 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
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119 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
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85 views

Winding number and homotopy

Given two maps $f,g : S^1 \rightarrow S^1$, I want to show that if they have the same winding number, then there is a homotopy between them. Well, we know that we can write them as $f(\exp(2 \pi i ...
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Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
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third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$

It is known that the third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$. Each colouring $C$ of a knot diagram by a dihederal quandle $R_3$ induces a homomorphism $f:C \rightarrow ...
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99 views

Comparing the torus and a small wedge product, and their universal coverings

I have the following problem: Let $A = S^1 \times S^1$ and $B = S^1 \vee S^1 \vee S^2$. Compute their universal coverings. Prove that $A$ and $B$ have isomorphic homology groups for any $n \in ...
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The vanishing (?) cohomology of the Milnor fiber

Setup. Say we have a germ of a holomorphic function $f:(\mathbb C^{n+1},0)\to (\mathbb C,0)$ with a critical point at the origin. There is an $\epsilon>0$ small enough so that $f$ becomes a ...
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45 views

Are 4-dimensional mapping tori always spin?

We know that all compact orientable manifolds of dimension 3 are spin. In 4 dimensions, $CP^2$ is not spin. I would like to ask if all 4-dimensional compact orientable mapping tori are spin?
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The Morse complex of a manifold with boundary

For a smooth manifold with boundary $M$ and $\partial M = V_+ \cup V_-$ two disjoint sets of boundary components, one usually defines the Morse complex of $M$ using a Morse-Smale pair $(f,X)$ such ...
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“Naive” smash products for spectra

Suppose I work in the completeley naive homotopy category of spectra, by which I mean sequences $E = (E_n)_{n = 0, 1, \dots}$ together with maps $\sigma_{E,n}: S^1 \wedge E_n \to E_{n+1}.$ We might ...
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112 views

Does the pullback of a covering space correspond to the pullback of the corresponding representation of $\pi_1$?

In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map ...
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115 views

Cross product of cohomology classes: intuition

Let $X$ and $Y$ be topological spaces and consider cohomology over a ring $R.$ Hatcher (in his standard Algebraic Topology text) defines the cross product of cohomology classes $$H^k(X) \times ...