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

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2
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3answers
65 views

What is the Quotient on the Coproduct in Adjunction Spaces

Can someone please provide a detailed explanation of the equivalence relation used to construct adjunction spaces from the topological coproduct? In particular, most sources talk about "identifying an ...
10
votes
0answers
44 views

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of ...
1
vote
0answers
15 views

Euler Integral of a self-overlapping tube with a cusp singularity

I am studying in depth the following paper on Euler calculus applied to target enumeration: https://www.math.upenn.edu/~ghrist/preprints/eulerenumerationpart1.pdf Within this paper there is an ...
6
votes
0answers
64 views
+50

Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?

Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h \in H\} \subset X;$$$X^H$ is the $H$-fixed point subspace of $X$. ...
3
votes
0answers
46 views

Computational Topology Codes

I am working on a project with a PI that thinks could be solved with computational topology tools. For this project, we will be looking at the persistent homology of objects in 3D images. I tried ...
4
votes
1answer
97 views

rationalization space

let $X$ be a topological space and $X_\mathbb Q$ its rationalization. 1) what is the rationalization of $X_\mathbb Q$, is it $X_\mathbb Q$ itself? 2) if $X$ is a CW complex, does that imply ...
0
votes
1answer
88 views

Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a ...
1
vote
0answers
58 views

Application of Morse Inequalities

I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ...
0
votes
0answers
43 views

Homotopic family of curves

I stumbled over the following question. Imagine we have a two homotopic curves on the sphere $\mathbb{S}^1$ namely $\gamma_1,\gamma_2$. Then we can write them as $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
3
votes
2answers
64 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
2
votes
0answers
23 views

The homology groups of an infinite product of spaces

Suppose $I$ is some index set and let $\{X_i \}_{i\in I}$ be a collection of topological spaces (as nice as you like them to be). What is known about the (say) singular homology of $X := \prod_{i\in ...
10
votes
0answers
129 views

Prove Euler characteristic is a homotopy invariant without using homology theory

I was flipping through May's Concise Course in Algebraic Topology and found the following question on page 82. Think about proving from what we have done so far that $\chi(X)$ depends only on ...
10
votes
2answers
395 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
1
vote
1answer
60 views

Action of $S^1$ on homotopy groups of an $S^1$-space

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed ...
6
votes
3answers
202 views

How can one compare these two 4-manifolds

We would like to compare the following two real 4 dimensional manifolds: 1)$M$=The tangent bundle of $S^{2}$ 2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$ ...
2
votes
2answers
66 views

Sum of Betti numbers of a degree $d$ hypersurface of $\mathbb{P}^n_{\mathbb{C}}$

Let $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ be the zero set of a homogeneous polynomial of degree $d$. Assume that $X$ is smooth. Is there a way to find the sum of the Betti numbers of $X$ (i.e. the ...
6
votes
1answer
87 views

Fundamental group of Antoine's necklace

Let $A \subset \mathbb{R}^3$ denote Antoine's necklace. It is well-known that $A$ is a Cantor space and that $\mathbb{R}^3 \backslash A$ is not simply connected. Futhermore, $\pi_1(\mathbb{R}^3 ...
4
votes
0answers
45 views

Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
2
votes
1answer
43 views

Question about maps to $K(G,1)$

I have an unfortunately basic confusion about two results in Hatcher's Algebraic Topology. Let $G$ be an abelian group. Theorem 4.57 specialized to $n=1$ says that there is a bijection $$\langle X, ...
15
votes
1answer
398 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
1
vote
2answers
69 views

Attaching maps for the CW-decomposition of the 3-torus

I want to calculate the homology of the $3$-torus via cellular homology. I figured out a CW-decomposition of the $3$-torus: $1$ $0$-cell, $3$ $1$-cell, $3$ $2$-cell, $1$ $3$-cell. So the chain complex ...
0
votes
1answer
40 views

Normality of a covering space.

I am having some trouble trying to do problem 1.3.24 in Hatcher's Algebraic Topology: Given a covering space action of a group G on a path-connected, locally path-connected space X, then each ...
2
votes
0answers
20 views

Properties of spectra

I've been trying to understand spectra better, and I was wondering what would be a good source to learn from. My issue with the classical treatments (e.g., Adams, Switzer) is that people keep telling ...
51
votes
3answers
899 views

Topological spaces admitting an averaging function

Let $M$ be a topological space. Define an averaging function as a continuous map $f:M \times M \to M$ which satisfies $f(a,b) = f(b,a)$ for all $a,b \in M$ and $f(a,a) = a$ for all $a \in M$. These ...
1
vote
0answers
25 views

Reference request about Thm which use Transversality to compute Homotopy Groups [duplicate]

I'm following the following notes, and my attention was caught by Theorem $1.1.4$. I am unable to find any reference of the proof. Could you suggest me some books in which there is a proof of this ...
2
votes
1answer
45 views

Todd genus as homomorphism from complex cobordism

It's well-known that the Todd genus/arithmetic genus $\chi(\mathcal{O}_X)$ (or probably preferably $\int_X \text{td}(T_X)$ so as to define it in purely terms of the complex structure) is a genus in ...
1
vote
1answer
56 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
1
vote
1answer
48 views

The cone of a topological space is contractible (why is the homotopy well defined?)

If $X$ is a topological space, define ${\rm Cil}(X) = X \times I$ the cylinder over $X$, and the cone over $X$, $\operatorname{Con}(X) = \operatorname{Cil}(X)/{\sim}$ the quotient by saying that ...
4
votes
1answer
38 views

Ways to link the unknot to a pole

Is there a way to show that the following ways of linking an unknot to an infinite horizontal pole are inequivalent? Perhaps the Wirtinger presentation would work, but I am not sure because of the ...
1
vote
1answer
29 views

What is the Wrong in this Triangulation of the Torus

On pg 133 of Roman's Introduction to Algebraic Topology it is stated that one requires at least 14 triangles in any triangulation of the torus. Admittedly, I do not have a very good understanding of ...
0
votes
0answers
53 views

Trying to prove that a continuous function $\,f: \mathbb{D}^2 \rightarrow \mathbb{R}^2$ has a zero using the fundamental group?

Prove: For a continuous $\;f: \mathbb{D}^2 \rightarrow \mathbb{R}^2,\;$ $\,f\left(\omega,0\right) \notin f\left(S^1\right),\,$ given a non constant function $\left(f_{\mid S^1}\right)_{\ast} : ...
44
votes
7answers
8k views

Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall ...
1
vote
1answer
30 views

Trouble With a Triangulation of the Torus

On pg. 133 of Rotman's Introduction to Algebraic Topology, we have a figure which claims to be a triangulation of the torus. Now a triangulation of a topological space is defined as Definition. ...
1
vote
1answer
24 views

Homology groups of $D^2\times S^1$, full torus

I know what are the homology groups of a torus $T=S^1\times S^1$, in sense that $$\tilde{H}_1(T)=\mathbb{Z}^2,H_2(T)=\mathbb{Z}$$ but I wonder what happens if we fill it. What are the homology groups ...
0
votes
1answer
14 views

Interior of simplex that is a proper face not open?

If a simplex $\sigma$ is a proper face of another simplex $\tau$, why is its interior not open in $\tau$? I can't seem to understand as, let's take $\sigma=ab$, a line segment that is a proper face ...
1
vote
2answers
64 views

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial.

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ is continuous and has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial. I'm not sure how the fact that there exists an ...
0
votes
1answer
39 views

Is my proof that the product of covering spaces is a covering space correct?

Let $p_1:\tilde X_1 \rightarrow X_1$ and $p_2:\tilde X_2 \rightarrow X_2$ be two covering spaces. Prove: $p = p_1 \times p_2:\tilde X_1 \times \tilde X_2 \rightarrow X_1 \times X_2$ is a covering ...
1
vote
1answer
49 views

If $r:X\to A$ is a Retraction, Then $H_n(X)\cong H_n(A)\oplus H_n(X,A)$

$\DeclareMathOperator{\im}{Im}$ Let $A$ be a subspace of a topological space $X$ such that there is a retraction $r:X\to A$ of $X$ onto $A$. Then $H_n(X)=H_n(A)\oplus H_n(X, A)$ for all $n$. ...
2
votes
2answers
420 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 ...
2
votes
2answers
76 views

In algebraic topology, for a function $f$ what does $f _\ast$ mean?

In algebraic topology, for a function $f$ what does $f_{ \ast}$ mean? I'm solving some exercises and this is something that's appearing, often relating to homotopic functions, and I'm not sure what ...
0
votes
1answer
26 views

Understanding the Product of $\Delta$-sets: $\Delta^1\times\Delta^1$

We learnt that a $\Delta$-set is said to be a sequence $K_\bullet$ of sets $\{K_n\}_{n\ge0}$ with defined "face" maps $d_i:K_{n+1}\rightarrow K_n$ for $0\le i\le n$ satisfying $d_id_j=d_{j-1}d_i$ ...
3
votes
1answer
401 views

Tangent bundle of Grassmann manifold

I have to prove that the tangent bundle of Grassmann manifold $G_n(\mathbb{R}^{n+h})$ is isomorphic to $\operatorname{Hom}(\gamma^n(\mathbb{R}^{n+k}),\gamma^\perp)$, with $\gamma^{\perp}$ is the ...
1
vote
2answers
31 views

Question of maps in Mayer-Vietoris sequence

We obtain MV-seq. from short exact sequence $$ 0\to C_n(A\cap B) \to C_n(A)\oplus C_n(B)\to C_n(A+B)\to 0 $$ So map i wonder that map $H_n(A\cap B)\to H_n(A)\oplus H_n(B)$ maps $[a]$ to $([a],[-a])$. ...
4
votes
1answer
40 views

Hirzebruch's $L$-polynomial and $\mathbb{C}P^n$

Hirzebruch's $L$-polynomial is the formal power series \begin{equation} L(x) = \frac{x}{\tanh x} = 1 + \frac{x^2}{3} + \cdots \end{equation} This defines a multiplicative sequence and a genus $L(M)$ ...
2
votes
1answer
66 views

The sphere $S^2$ is not contractible

I heard that in topology the sphere $S^2$ cannot be continuously deformed to a point, i.e. $S^2$ is not contractible. Sorry for my ignorance, but I really don't get it. Can't we just push all the ...
49
votes
3answers
1k views

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
12
votes
3answers
1k views

Toy sheaf cohomology computation

I asked this question a while back on MO : One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$) I am curious, as a sort of followup if ...
7
votes
1answer
342 views

Varying definitions of cohomology

So I know that given a chain complex we can define the $d$-th cohomology by taking $\ker{d}/\mathrm{im}_{d+1}$. But I don't know how this corresponds to the idea of holes in topological spaces (maybe ...
0
votes
0answers
22 views
2
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0answers
33 views

Why is the singular $p$-simplex called singular?

I am learning homology. I cannot see in which sense the singular $p$-simplex is singular. It seems quite regular. Any idea?