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

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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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26 views

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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25 views

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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44 views

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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35 views

Cohomology of $S^2\times S^2/\mathbb{Z}_2$

The product of two spheres admits a diagonal $\mathbb{Z}_2$-action, $(x,y)\mapsto (-x,-y)$. I'm trying to compute the integral singular cohomology ring of the orbit space $X$ of this action. $X$ is ...
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34 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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38 views

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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47 views

Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then ...
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89 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathcal{Top}$, $k$-closed subset $Y\subset X$, means $u^{-1}(Y)$ is closed in $C$ for any $u: ...
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36 views

“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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39 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
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57 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
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50 views

Singular cohomology with compact support

If $X$ is a locally compact Hausdorff space, then for any $n \geq0$ is $H_c^n(X) \cong {\tilde H^n}({X^ + })$? ($H_c^n(X)$ is the Singular cohomology with compact support and $X^+$ is the one-point ...
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38 views

Why is this Milnor fiber homeomorphic to a cylinder?

Let $f:(\mathbb C^2,0)\to (\mathbb C,0)$ be a holomorphic function with a critical point at the origin. Let us denote by $X_0$ the fiber $f^{-1}(0)$, in which, as we said, the point $0\in X_0$ is a ...
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50 views

homomorphism of fundamental groups induced by a map of two surfaces

I am trying to find another proof of the following theorem Theorem Let $X$ and $Y$ be two compact surfaces of genus greater than $2$. Then every homomorphism $π_1(X,x_0)→π_1(Y,y_0)$ is induced by a ...
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51 views

Can injective and projective model structures be the same ?

Is there a non trivial example where injective and projective model structures coincide ? By non trivial I mean an example of a functor category on a non discrete base category where I guess ...
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54 views

English edition of Vol 9 of Dieudonné's Foundations of Modern Analysis?

I have found the first 8 volumes of Dieudonné's Foundations of Modern Analysis in English translation, but I'm having difficulty locating volume 9. I have searched the catalogues of numerous libraries ...
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121 views

Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
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42 views

Path-homotopic definition.

Given two paths $f,g: [0,1] \mapsto X $ Then their product is defined by $f\cdot g := \begin{cases} f(2t) , \space 0\leq t \leq \frac{1}{2}\\ g(2t-1) ,\space \frac{1}{2} \leq t \leq 1\\ \end{cases} ...
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137 views

Adjointness of homological and cohomological transfer maps

Suppose $p: \tilde X \to X$ is a normal covering space with deck group $\Gamma$. Then there are transfer maps $$ \tau^*: H^*(\tilde X, \mathbb Q) \to H^*(X, \mathbb Q) $$ and $$ \tau_*: H_*(X,\mathbb ...
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37 views

Reference request: where can I find illustrative, concrete examples of the use of the Eilenberg–Moore spectral sequence?

Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle $S^3 \to ...
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180 views

Are Infinite Simplicial Complexes Hausdorff?

This is my definition of an Infinite Simplicial Complex: Let $V$ be an (infinite) set. Let $\Sigma$ be a collection of finite nonempty subsets of $V$ such that: $$ \forall v \in V \space \space ...
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66 views

Integral Homology of $BU$

We know that the integral cohomology of $BU$, $H^*(BU) = Z[c_1,c_2,...]$ where $|c_i| = 2i$ is the $i$th Chern class, with coproduct $\Delta c_i = \Sigma c_j\otimes c_{i-j}$. And at almost ...
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33 views

Is a sufficiently nice simple curve which is nulhomotopic the boundary of a surface?

This is a follow up to Is a simple curve which is nulhomotopic the boundary of a surface?. There, I asked whether, given a simple curve $C$ in an open subset $U$ of $\mathbb R^3$ which is nulhomotopic ...
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36 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
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53 views

The groups $[\Sigma^nX,Y]$ versus the homotopy groups

If $X,Y$ are pointed spaces, denote by $[X,Y]$ the pointed homotopy classes of pointed maps $X\to Y$. The sets $[\Sigma^nX,Y]$ actually have the structure of a group for $n\geq 1$. Here $\Sigma$ ...
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33 views

Total space of a geometric vector bundle

Suppose that $X$ is a scheme with a $B$-action, and that $\lambda: B \to \mathbb{C}^*$ is a character of $B$. We then have a geometric vector bundle $\pi: X \times^B k \to X/B$, where $B$ acts on $X ...
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43 views

differential forms on covering spaces

I seem to be under the impression that if $p:A \to B$ is a regular covering (of smooth manifolds) with $\alpha\in \Omega^k(A)$; there exists $b\in \Omega^k(B)$ such that $\alpha= p^*\beta$ if and only ...
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36 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 ...
3
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71 views

Classification of circle bundles over a 2-manifold with boundary

I want to understand and try to give a proof of the following claim: Let $B$ be a compact, connected topological $2$-manifold (surface) with nonempty boundary, then the $S^1$-bundles over $B$ with ...
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62 views

Homological Interpretation of the intersection number

Let $M^r$ and $N^s$ be smooth submanifolds of the smooth manifold $V^{r+s}$ and $M,N,V$ compact, connected and orientable. The Thom-Isomorphism together with the Tubular Neighbourhood Theorem gives me ...
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70 views

Products of homology groups

In the topology course I attend we work with the following, rather unusual definition of homology groups: For topological spaces $X,Y$ define the presheaf $\mathcal{F}_Y$ on $X$ as follows: Let ...
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71 views

Mayer-Vietoris of pair (X,C)

I would like to know if i can use Mayer-Vietoris with this form: Let X be a topological space and A, B be two subspaces whose interiors cover X and $C\subset A\cap B$. We get the exact sequence ...
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110 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
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46 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
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64 views

Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i ...
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97 views

Discrete Closed Subgroup H of a Simply Connected Topological Group G isomorphic to fundamental group of G / H.

A problem in Rotman's Algebraic Topology is as follows: Given a simply connected topological group G with a closed discete normal subgroup H, show that $\pi_1(G / H) \cong H$. I believe I have this ...
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94 views

Smash product of compact spaces

In the topology book I'm reading I found the following statement: The "smash product" (of two pointed spaces) is defined as $X \bigwedge Y=X \times Y/(X \times \lbrace*\rbrace \bigcup Y \times ...
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45 views

Serre spectral sequence and locally constant coefficients

I have a brief question - In the Serre Spectral sequence for a fibration $$F \rightarrow E \rightarrow B$$ one can require, to avoid using local system of coefficients, that the action $\pi_1(B)$ on ...
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167 views

Galois correspondence of covering spaces of spaces not necessarily semilocally simply-connected

I've been trying to solve the following exercise (1.3.24) from Hatcher's Algebraic Topology: Given a covering space action of a group $G$ on a path-connected, locally path-connected space $X$, ...
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207 views

Want to show two maps are homotopic

I am trying to solve the following problem but so far I cannot do it. Let $X$ be a connected CW-space such that its homotopy group is 0 except for the fundamental group. Let $M$ be a closed manifold ...
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28 views

Relative Hopf degree theorem

If $f,g$ are two maps from $(D^n,S^{n-1})$ to $(D^n,S^{n-1})$ such that they have the same degree, that is $f_*[\mu]=g_*[\mu]$ where $[\mu]$ is a generator of $H_n(D^n,S^{n-1})$, then can we find a ...
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53 views

Understanding J homomorphism

I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism. It's clear that an element of $\pi_r(SO(q)$ ...
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76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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79 views

Euler characteristic of affine space

Sorry for the trivial question.. but what is the (topological) Euler characteristic of $\mathbb{A}^n$? Also, is there a genus-degree formula for affine curves similar to $g={d-1\choose 2}$ for smooth ...
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63 views

What is a $c_1$-map for Riemann-Roch theorems?

Atiyah and Hirzebruch define a $c_1$-map to spell out Riemann-Roch theorem for (compact and connected) smooth manifolds. The definition is following: a map $f:Y \to X$ is called a $c_1$-map if we are ...
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86 views

What big families of theories/structures are there?

To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families? universal algebra: classes of objects with arbitrary ...
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96 views

Homotopy of a CW complex

I have a CW complex constructed as follows: (The circle and the rectangles are 2-cells, different 1-cells are denoted by different colors, and there is one 0-cell). We can see it as gluing two Klein ...
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66 views

Proof that continuous $f: S^1 \rightarrow S^1$ satisfying $f(x) = -f(-x)$ represents an injection on homology

I'm looking to prove that continuous functions $f: S^1 \rightarrow S^1$ satisfying $f(x) = -f(-x)$ for all $x \in S^1$ represent injections on homology. I'm trying to prove this fact on the way to ...
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110 views

Simplicial homology for n-simplex

I've just started to study homology theory. And I'm trying to calculate all $H_n(\Delta_N)$ for some $N$. I know that the number of $m$-simplex in $N$-simlex is $b_{N,m}={N+1 \choose ...