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
30 views

Can every basic concept of fundamental group be generalized to homotopy group?

I'm taking (undergraduate) algebraic topology this year and I have learned some basic concepts in this subject. I found this subject interesting, but it seems like the usefulness of fundamental groups ...
0
votes
0answers
13 views

Positive definite functions defined on the embedding of a planar graph in the plane

By way of motivation: Let $f:[0,1]\rightarrow\mathbb{R}^2$ and $g:\mathbb{S}_1\rightarrow\mathbb{R}^2$ be continuous injections, where $\mathbb{S}_1$ is the circle ($\mathbb{R}/\mathbb{Z}$). Then, ...
2
votes
1answer
78 views

How to prove that $CP^4$ cannot be immersed in $R^{11}$

Please let me know how to prove that $CP^4$ cannot be immersed in $R^{11}$. I know a proof using an integrality theorem for differentiable manifolds but I want to know if a more direct and simple ...
3
votes
0answers
15 views

Postnikov towers for non-CW spaces

In the literature, Postnikov systems seem to be defined always in the setting of CW complexes. Looking at the proofs, it is not clear to me, why this assumption should be necessary. Question: Does ...
-1
votes
0answers
20 views

cohomology homomorphism induced by classifying map [on hold]

(1). Prove that there exists a principal $Sp(1)(\cong S^3)$-bundle over $\mathbb{C}P^\infty$, denoted as $Sp(1)\to E\to \mathbb{C}P^\infty$, such that $E\simeq S^2$. (2). The universal bundle ...
0
votes
0answers
14 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow ...
0
votes
0answers
17 views

fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
5
votes
1answer
28 views

$H_2(M)$ is free abelian for any simply connected $4$-manifold

In Naber's book "Topology, Geometry and Gauge Fields. Foundations", it is stated that for each $4$-manifold $M$ which is smooth, closed, connected and simply connected we have $H_0(M) = H_4(M)= ...
3
votes
0answers
41 views
+50

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
1
vote
0answers
93 views
+50

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
0
votes
0answers
15 views

Does Brouwer's theorem holds in every closed path-connected subset of $\mathbb{R}^2$

Here is an assignment my professor handed out: Let $X$ be some object crushed from a closed ball $B^2$. Let $f:X\rightarrow X$ be a continuous function. Then, there exists $x$ such that ...
2
votes
1answer
275 views

K-fold covering

I'd like some help with this homework: Let $p: E\to B$ be a covering map; let $B$ connected. Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0 \in B$, then $p^{-1}(b)$ has $k$ elements for ...
0
votes
1answer
31 views

mapping cylinder contractible iff Hn(f):Hn(X)->Hn(Y) is an isomorphism

The mapping cylinder will be defined as $Z_f=X\times[0,1]\coprod Y/\sim$, where $\sim$ is defined by $(x,1)\sim f(x)$. Let $f:X\to Y$ a continuous map between topological spaces and the map ...
3
votes
0answers
95 views

What is a good, hi-tech textbook on complex analysis?

I am looking for an introductory textbook for Complex Analysis that is hi-tech. All the books I have looked at suffer from the same problem; they're only assuming that the reader is familiar with is ...
1
vote
1answer
14 views

Product of infinite covering maps

If $p_i:E_i\rightarrow B_i, \ i\in I$ are covering maps, then is it true that $$\prod_{i\in I} p_i:\prod_{i\in I}E_i \rightarrow \prod_{i\in I}B_i$$ is a covering map ? It is true if $\mbox{card}\ I ...
3
votes
1answer
95 views

Are cofibrant/fibrant replacements homotopically unique?

Let $\mathcal{M}$ be a model category in the old sense, i.e. with factorisations that are not necessarily functorial, and let $X$ be an object in $\mathcal{M}$. The category of cofibrant replacements ...
1
vote
2answers
154 views

Proof that a continuous map to $S^n$ whose image is a proper subset of $S^n$ is null-homotopic

I am attempting to prove the following: If $g:X \to S^n$, $n \ge 1$, is a continuous map whose image $g(X)$ is a proper subset of $S^n$, then $g$ is null-homotopic. Just before this I proved that if ...
3
votes
1answer
85 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
2
votes
1answer
24 views

$t$-adic topology (on $\mathbb F_p(1/t)$)

Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic topology. The $t$-adic ...
1
vote
1answer
25 views

How do I conclude that $|\pi_1(X,x_0):p_*(\pi_1(C,c_0))|$ is the number of sheets of $p$?

Here is a theorem in Hatcher's algebraic topology. (Hatcher-Algebraic Topology p.61) Let $(X,x_0),(C,c_0)$ be topological spaces and $p:(C,c_0)\rightarrow (X,x_0)$ be a covering map. If ...
0
votes
0answers
8 views

Is the range of a covering map normal subgroup of codamin?

Let $(C,c_0),(X,x_0)$ be topological spaces. Let $p:(C,c_0)\rightarrow (X,x_0)$ be a covering map. Let $p_*:\pi_1(C,c_0)\rightarrow \pi_1(X,x_0)$ be the induced homomorphism by $p$. ...
0
votes
0answers
34 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
0
votes
0answers
43 views

How do I show that $\mathbb{R}^n$ is simply-connected?

I have shown that $\mathbb{R}$ is simply connected by reparametrization. However, how do I show that $\mathbb{R}^n$ is general?
1
vote
1answer
24 views

Why does a covering map has the injective induced homomorphism?

Below is how I tried: Let $p:(C,c_0)\rightarrow (X,x_0)$ be a covering map. Let $[\gamma]\in \ker(p_*)$ Let $e_X,e_C$ be the constant loops at $x_0,c_0$ respectively. Then $[e_X]=[p\circ \gamma]$. ...
2
votes
1answer
52 views

Acyclic chain complex and contracting chain homotopy

Let $R$ be a Ring and $(C_k, d_k)_{k\geq0}$ a acyclic chain complex of free modules, meaning $im(d_{k+1})=\ker(d_k)$ for all $k$. I want to show that there is a family of R-module-homomorphisms ...
1
vote
1answer
52 views

Homomorphism extension between fundamental groups

I have the following problem. Let $X$ be a subspace of $\mathbb{R}^n$ and $Y$ some topological space. Also, let $\psi: (X,x_0) \to (Y,y_0)$ be a continuous map. If there exists a continuous extension ...
1
vote
1answer
41 views

algebra of polynomials of non-commuting variables

In Hatcher's book algebraic topology page 496-497: the Steenrod algebra $\mathcal{A}_2$ is the algebra over $\mathbb{Z}_2$ formed by polynomials of non-commuting variables $Sq^1,Sq^2,\cdots$ modulo ...
2
votes
2answers
46 views

path connected subspace $A$ of $X$, $ i:A\to X$ inclusion. Why is the induced map of $i$ on homology injective?

X is a topological space and $A\subset X$ is a path-connected subspace of X and $i:A\to X$ is the inclusion. I want to know, why the induced map of i on singular homology of dimension zero over ...
0
votes
2answers
43 views

Group with topology which is not topological group

What will example of a group G with topology such that f: G to G such that f(x) = -x and g: G * G to G such that g((x,y)) = x * y (where * is binary operation on G) both are not continuous.
3
votes
2answers
66 views

Are there any disadvantages to working in the category of k-spaces as opposed to Top?

Unlike the category Top of topological spaces with continuous maps as the arrows, the full subcategory of compactly generated spaces (k-spaces) is Cartesian closed. It seems like a very nice ...
1
vote
0answers
32 views

Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated?

Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated ? Or Are there some reference books especially on differential geometry and ...
1
vote
2answers
22 views

How can “homotopy lifting theorem” be applied to prove this theorem?

Homotopy lifting theorem Let $p:C\rightarrow X$ be a covering map. Let $F:Y\times[0,1]\rightarrow X$ be a continuous function. Let $f:Y\rightarrow C$ be a continuous function such ...
0
votes
3answers
62 views

Boundary of $\mathbb{R}^4$ and fundamental group of $\mathbb{R}^4/\mathbb{R}^2$

a) To what I know boundary makes no sense in open sets. Does it make any sense to talk about the boundary of $\mathbb{R}^4$? In physics they consider it when discussing wether the universe has a ...
3
votes
2answers
92 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
3
votes
2answers
268 views

Mapping the open ball to itself?

How to prove that there exists a continuous function $f:B^2 \to B^2$ without constant points? Here, $B^2$ is the unit open ball. I guess $f$ can be for example like this $f: re^{iax} \to re^{ibx} $ ...
2
votes
1answer
15 views

What is the definition of “sheet”?

Here is a definition for evenly covered set. Definition Let $C,X$ be topological spaces and $p:C\rightarrow X$ is a continuous function. Let $U$ be an open subset of $X$. Then $U$ ...
2
votes
1answer
32 views

Fixed point of a mapping

How to prove that every continuous $f:S^1 \to S^1$ such that $deg(f)\neq 1$ has a fixed point? One hint is that if $f(x)\neq x$ for any $x\in S^1$ then $f$ is homotopic to the antipodal map $a$ but I ...
3
votes
2answers
89 views

Circle to circle homotopic to the constant map?

How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$ (a) has a constant point and that (b) $f$ maps $x$ to its antipodal point $-x$?
0
votes
0answers
42 views

Geometric Meaning of Luna's Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
2
votes
1answer
74 views

$p:E\to B$ is fibration then $p^*:map(X,E)\to map(X,B)$ is fibration as well.

$p:E\to B$ is fibration then for $X$ being completely generated weakly Hausdorff space $p^*:map(X,E)\to map(X,B)$ is fibration as well. We'd like to show that for any $Y$ and continuous $f$ and ...
1
vote
1answer
27 views

Compute the map $H^*(CP^n; \mathbb{Z}) \rightarrow H^*(CP^n, \mathbb{Z})$

I'm trying to solve problem 3.2.6 in Hatcher. The problem is stated: Use cup products to compute the map $H^*(CP^n; \mathbb{Z}) \rightarrow H^*(CP^n, \mathbb{Z})$ induced by the map $CP^n ...
1
vote
0answers
15 views

How do i have this neighborhood in this argument?

(Hatcher- Algebraic topology) p.30 To prove (c) we will first construct a lift $\overline{F}:N\rightarrow \mathbb{R}$ for $N$ some neighborhood in $Y$ of a given point $y_0 \in Y$. Since $F$ is ...
1
vote
0answers
28 views

is configuration space an H-space?

Let $X$ be a manifold. Let $F(X,n)$ be the configuration space of order $n$. Let $B(X,n)=F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. Is $B(X,n)$ an H-space? Under what ...
0
votes
1answer
6 views

Formal sums vs. arbitrary sums for chain groups

We defined the $n$-chain group as follows, $$C_n(X) = \bigg\{ \sum_{v \in V} n_v [v_1, \dots, v_n] : n_v \in \mathbb{Z}, \hspace{2mm} n_v=0 \hspace{2mm} \text{for all but finitely many} \hspace{1mm} ...
0
votes
0answers
16 views

$\phi$ is a coboundary iff $\phi(f)$ depends only on the endpoints of $f$

I've proved the first direction but I'm having trouble proving the second direction. First direction: Let $\phi = \delta\psi$ for $\psi \in C^0(X;G)$. Then $\phi(f) = \delta\psi(f) = ...
0
votes
1answer
48 views

Proof of Jordan curve theorem

Is it possible for the following to be proof for Jordan curve theorem: Given the distance function on $\mathbb{R}^2$ ($d((x_1,y_1),(x_2,y_2))=\sqrt{ |x_2-x_1|^2 + |y_2-y_1|^2}$), and $\varepsilon ...
4
votes
1answer
63 views

Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?
1
vote
1answer
309 views

Klein bottle covered by the torus

Maybe this is an idiot question and I'm missing something very trivial. This question question was asked here before, but the answer (which apparently is equal to the one that I created) seems ...
1
vote
0answers
26 views

Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$

For $\phi \in C^1(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
0
votes
2answers
25 views

Fundamental Group of Circle Generator Textbook Typo?

I'm confused about what the generator is for the fundamental group of a circle at point $b_0$. That is, what is the generator for $\pi_1(S^1, b_0)$. Is it $e^{2\pi i (t_0 + t)}$ for $t \in [0, 1]$, ...