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

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2
votes
1answer
111 views

Refining homotopy commutative maps of spectra to maps of E_{\infty}-ring spectra

In Adams' blue book (page 54) we have a map in the homotopy category of ring spectra $f: MU \rightarrow K$ where $K$ is complex $K$-theory such that $g_*x^{MU} = (u^K)^{-1}x^K$ where $x^E$ denote ...
3
votes
2answers
61 views

vector bundles on $X \times S^2$ built via clutching functions

Let $X$ be a compact Hausdorff space and $E \to X$ a (complex) vector bundle over $X$. We can build a vector bundle on $X \times S^2$ by clutching functions, i.e an automorphism $f:E \times S^1 \to E ...
2
votes
2answers
56 views

Isomorphism of modules arising from algebraic topology

While studying for a course in algebraic topology, the following question popped out: Let $S,R$ be two commutative rings with unit, $A,B$ two $S$-modules, and assume that $R$ is also an ...
2
votes
1answer
29 views

The cone minus its apex deformation retracts onto its basis

Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$. My ...
11
votes
3answers
139 views

Fundamental group of the product of 3-tori minus the diagonal

I have a past qual question here: let $T^3 = S^1 \times S^1 \times S^1$ be the 3-torus, and let $\Delta = \{ (x,x) \in T^3 \times T^3 \colon x \in T^3 \}$ be the diagonal subspace. Compute $\pi_1(T^3 ...
-1
votes
0answers
19 views

AI / Machine Learning related to high/modern/front mathematics [on hold]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
1
vote
0answers
29 views

Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
0
votes
0answers
32 views

Why is $(X\times EG)/G\to X/G$ a fibration if $G$ acts freely on $X$?

Suppose that $G$ acts freely on $X$, and let $EG$ be a contractible space on which $G$ acts freely. According to many references, the projection $(X\times EG)/G\to X/G$ is a fibration. However, I ...
3
votes
1answer
38 views

Extending cellular maps between aspherical complexes

In a paper I read, the author seemed to use a property similar to: Let $X, Y$ be two aspherical CW-complexes and $f : X^{(2)} \to Y^{(2)}$ be a cellular map between their 2-skeletons. Then $f$ ...
2
votes
4answers
274 views

Should I read about Manifolds or Algebraic Topology?

I really enjoy doing maths and it fills quite a lot of my spare time. I'm starting my first year in the university on october and I probably won't have that much time for independent reading once ...
0
votes
1answer
33 views

What are the deck transformations of this threefold cover of the figure 8?

Hatcher lists some examples of covers of a figure 8 (page 58). One of them corresponds with the group with two generators $a$ and $b$ and the relations $a^2, b^2, aba^-1, bab^-1$. I thought ...
0
votes
2answers
33 views

Why is $H_k(X^n) = H_k(X)$, $k < n$, where $X^n$ is the $n$-skeleton of the CW-complex $X$?

Why is $H_k(X^n) = H_k(X)$, $k < n$, where $X^n$ is the $n$-skeleton of the CW-complex $X$? I am probably overlooking something trivial. I tried using the fact that $H_k(X^n,\emptyset)\cong ...
2
votes
0answers
39 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
83
votes
1answer
2k views

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
3
votes
0answers
42 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 ...
3
votes
1answer
38 views

Question about the proof of the universal coefficient theorem

When deriving the universal coefficient theorem, in class we proceeded as follows: We have the SES: $$0\to ...
3
votes
1answer
49 views

Eilenberg-Maclane space $K(G\rtimes H, 1)$ for a semi-direct product.

We know that $K(G\times H, 1)=K(G,1)\times K(H,1)$. Do we know something like this for a semi-direct product, where $K(G,1)$ denotes the Eilenberg-Maclane space.
1
vote
0answers
33 views

Homology group of the join

Prove that $$\tilde{H_n}(X*Y) \cong H_{n-1}(X \times Y/X \vee Y)$$ Firstly, I set $$A=X \times Y \times [0,1) / (x,y_0,0) \sim (x,y_1,0) $$ $$B=X \times Y \times (0,1] / (x_0,y,1) \sim (x_1,y,1) $$ ...
0
votes
1answer
70 views

Show that an inclusion is an isomorphism in homology

I'm struggling a bit with an exercise from a book, in a chapter about the Jordan-Brouwer separation theorem. It goes as follows: (note: $s_{n-1}$ is a topological space homeomorphic to ...
2
votes
2answers
48 views

Find a CW complex with prescribed homology groups

A past qual question asks to construct a CW-complex $X$ with $H_0(X) = \mathbb{Z}$, $H_5(X) = \mathbb{Z} \oplus \mathbb{Z}_6$, and $H_n(X) = 0$ for $n\not= 0, 5$. One can build a CW-complex $Y$ by ...
1
vote
1answer
33 views

Show this Map $f:M\rightarrow S^n$ is a Diffeomorphism

Let $M$ be a compact, connected smooth manifold of dimension $n>1$, and $f:M\rightarrow S^n$ a smooth map such that the differential $df_x:T_xM\rightarrow T_{f(x)}S^{n}$ has full rank $= n$ for all ...
2
votes
1answer
28 views

Homology groups of a simplicial complex

I have a question from a qualifying exam: let $X$ be the simplicial complex that consists of the 3-simplices $(v_1,v_2,v_3,v_4)$, $(v_3,v_4,v_5,v_6)$, $(v_1,v_2,v_5,v_6)$, where the $v_i$'s are all ...
0
votes
1answer
24 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
11
votes
5answers
993 views

Poincare Duality Reference

In Hatcher's "Algebraic Topology" in the Poincare Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's ...
5
votes
2answers
117 views

Is there a self-homeomorphism of the 2-sphere with exactly 3 fixed points?

I don't believe so, but I'm not sure how to prove it. The Lefschetz-Hopf theorem says in this case that the sum of the fixed point indices is 0 or 2 (since our map is a self-homeomorphism). My ...
0
votes
1answer
56 views

A detail in the proof of Jordan's theorem

The (usual) proof with homology first inductively shows that complements of embedded disks are acyclic. In doing so, Mayer-Vietoris is applied, and this assumes that their complements in the sphere ...
1
vote
1answer
30 views

Number of connected components of the complement of a closed curve.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 ...
3
votes
1answer
65 views

Algebraic question (In Hatcher's book, exercise: 1.1.16-c:)

In Hatcher's book, exercise: 1.1.16-c: Show that there are no retractions $r :X \rightarrow A$ in the following cases: (c) $X = S_1 × D_2$ and A the circle shown in the figure. Page 39 By ...
3
votes
1answer
28 views

Two definitions of framed manifolds/cobordism

One notion of framed cobordism is that $\Omega_{fr}^n(X)$ is a certain equivalence class of $n$-manifolds $M^n$ in $X$ with a trivialization of the normal bundle $N M^n\subseteq TX$. For compact ...
8
votes
2answers
288 views

CW complexes and manifolds

What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework... ...
1
vote
2answers
102 views

Representation of nullhomologous loop on compact surface as a product of commutators.

Why this sentence is true?: Assume that $M$ is compact surface and $f: S^1 \to M$ is nullhomologous and without selfintersections. Letting $g$ be the genus and $b$ the number of boundary components ...
4
votes
3answers
387 views

Are these two spaces homotopy equivalent?

Let $X$ be the $2$-sphere with two pairs of points identified, say $(1,0,0) \sim (-1,0,0)$ and $(0,1,0) \sim (0,-1,0)$. Write $Y$ for the wedge sum of two circles with a $2$-sphere: if it matters, the ...
6
votes
2answers
99 views

Is this Space Homotopy Equivalent to $S^2$

Let $X$ be the space $S^1$ with two $2$-cells attached via maps of relatively prime degrees. This space is simply connected and has the homology of $S^2$, but is it homotopy equivalent to $S^2$?
3
votes
1answer
38 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
1
vote
2answers
63 views

How to call two subsets that can be deformed into each other?

Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$? $A \sim B :\Leftrightarrow \exists \text{ continuous } ...
2
votes
0answers
33 views

Cellular structure on a manifold [duplicate]

Is it always possible to put a cell structure on a manifold? in other words is it possible to decompose a manifold as a CW complex? I know that by Morse theory we always have a handle decomposition of ...
2
votes
1answer
59 views

What are other examples of characteristic numbers?

Be warned, this may be a ridiculous question. I understand characteristic classes of principal $G$-bundles (and associated vector bundles) over a space $X$ arise from the classifying maps $f\colon X ...
5
votes
1answer
74 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
8
votes
2answers
212 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
3
votes
1answer
51 views

Trivial Cohomology Group->Lower-Dimensional Homotopy?

Calculating the (de-Rham) cohomology of a tee connector (Picture), I got $H^0=R,H^1=R^2,H^2=0$. Furthermore, just from looking at it, I assume the tee connector is homotopic to a circle with an arc ...
1
vote
1answer
21 views

A further question on reparametrization.

Hatcher contains the following paragraph: Define a reparametrization of a path $f$ to be composition $f\psi$ where $\psi:I\to I$ is any continuous map such that $\psi(0)=0$ and $\psi(1)=1$. ...
3
votes
2answers
114 views

the mapping class group of the disk is trivial proof

Proof : Identify $D^2$ with the closed unit disk in $\mathbb{R}^2$. Let $\phi : D^2 \rightarrow D^2$ be a homeomorphism with $\phi_{\partial D^2}$ equal to the identity. We define, $F(x,t) = ...
2
votes
1answer
56 views

Proof of an algebraic topological lemma

I have been given the following result without proof, so I would like to show it is true: Let $I=[0,1]$, then: $$H^\bullet(I,\partial I;R)\cong H^\bullet(I/\partial I,*;R)\cong ...
9
votes
1answer
185 views

Relative de Rham Cohomology is Homotopy Invariant

Suppose $ f:N\rightarrow M$ is a smooth map between two manifolds. Relative de Rham cohomology is defined through the complex $ \Omega^{q}(f)=\Omega^{q}(M)\oplus\Omega^{q-1}(N)$ with ...
1
vote
2answers
49 views

$H_n(\mathbb{R}P^4 \times S^1)$

I have been trying to compute the homology of $\mathbb{R}P^4 \times S^1$ by using cellular homology. Nevertheless, I cannot see what the attaching maps are.
3
votes
1answer
90 views

Complex (topological) K-theory of $S \Sigma$ for a surface $\Sigma$.

In the following question here: What's the K-group of a surface? The $K$-theory of a compact orientable surface is computed. I was curious if it was also possible to compute the "higher" ...
0
votes
0answers
29 views

How closely is algebraic topology related to mathematical physics? [closed]

I have heard that research in string theory uses mostly pure math . Sounds fascinating....
1
vote
0answers
39 views

Homotopy groups of unitary groups

in this paper I found some explicit generators of homotopy groups of unitary groups, for example $\pi_3[SU_2]$: $\begin{bmatrix}z_1\\z_2\end{bmatrix}$$\rightarrow $$\begin{bmatrix}z_1 ...
7
votes
2answers
102 views

Long exact sequence of a fibration, center

Let $p:E \rightarrow B$ be a fibration with fiber $F$ . Associated to this we have a long exact sequence $$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow ...
1
vote
0answers
24 views

How to show that the local homology at a point on the boundary of the half-plane is zero?

Let $\mathbb{H}^n = \{x = (x_1,\dots,x_n) \in \mathbb{R}^n \mid x_n \geq 0 \}$. i.e. the half plane. Let $y \in \mathbb{R}^{n-1} \times \{0\}$ be a point on the boundary. How do I show that ...