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

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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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30 views

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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32 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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37 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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38 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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41 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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32 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 ...
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43 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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36 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 ...
3
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36 views

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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46 views

definitions of various spectra: $E^X$ and $E \wedge \Sigma^\infty X$

Let $E=\{E_n\}$ be a spectrum given by a sequence of pointed CW complexes $E_n$ and inclusions $\Sigma E_n \to E_{n+1}$. Let $X$ a pointed CW complex. I had a few very naive questions I had while ...
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50 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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48 views

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
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36 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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58 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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47 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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58 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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38 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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23 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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45 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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68 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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29 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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31 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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53 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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41 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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36 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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43 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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51 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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108 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 $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in ...
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41 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 ...
3
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57 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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40 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 ...
3
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58 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 ...
3
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59 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 ...
3
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54 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 ...
3
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60 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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44 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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40 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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198 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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78 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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39 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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57 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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36 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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46 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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38 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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94 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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67 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 ...