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

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5
votes
1answer
29 views

$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)$

For Abelian groups $\pi$ and $\rho$, what is the easiest way to see that $$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)?$$My idea is the use the natural isomorphism$$[X, K(\rho, n)] \cong ...
3
votes
1answer
46 views

$\pi_7(S^4)$ contains an element of infinite order.

Show that $\pi_7(S^4)$ contains an element of infinite order. Now, I know that I should probably use the Hopf bundle here somewhere. I also know that $\pi_3(S^7) = 0$. But I am stuck. Can anyone ...
1
vote
0answers
17 views

Fibration implies inclusion is based homotopy equivalence?

If $p: E \to B$ is a fibration, does it follow that the inclusion$$\phi: p^{-1}(*) \to Fp$$specified by $\phi(e) = (e, c_*)$ is a based homotopy equivalence?
3
votes
0answers
15 views

Why study the p-completions of a space?

Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or ...
2
votes
1answer
23 views

Coproduct of the homology coalgebra of the sphere

Let $S^m$ be the $m$-sphere and $H_*(S^m)$ be the homology coalgebra with field coefficient. Then what is the coproduct of $ H_*(S^m) $? For $x$ the generator of $H_*(S^m)$, does $$ \Delta_*x=0? $$ ...
3
votes
1answer
45 views

Geometric Interpretation of Chain Homotpy

Let $X$ and $Y$ be topological spaces. Two maps $f,g:X\to Y$ are said to be chain homotopic if for each $n$ we have a map $T_n:C_n(X)\to C_{n+1}(Y)$ such that ...
9
votes
1answer
67 views

Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

There is a short argument using Zorn's lemma and the compactness of $[0,1]$, that shows every manifold must have maximal open simply connected subspaces. However, I am wondering if it is necessarily ...
0
votes
0answers
46 views

Covering Space Question

I recently encountered the following: Let $p:(E, e_0) \to (B, b_0)$ be a covering map. Assume that $p_∗(\pi_1(E, e_0)) \subseteq \pi_1(B, b_0)$ is a normal subgroup. If $e_1\in p^{−1}(\{b_0\})$, then ...
0
votes
0answers
27 views

inclusion of homotopy fiber and induced map on homology group

Given a fibration $F \to E \to B$, under what circumstances does the inclusion of the homotopy fiber into $E$, $F \to E$, induce injections on homology? The specific case I'm dealing with involves the ...
0
votes
0answers
32 views

Can a linear projection of spheres be a torus?

Assume that we have two disjoint subsets $A_1, A_2 \in \mathbb{RP}^4$ that are both homeomorphic to the sphere $S^2$. Let $\pi$ be the linear projection with centre a point that does not lie on $A_1$ ...
0
votes
0answers
27 views

Extensions and Pushouts using an exact sequence of sets

This might seem a strange way of doing things, that is, inventing a possible example (according to comments, there is no such thing as an exact sequence of sets), but let us try to make one for ...
0
votes
1answer
22 views

How many Vertices, Edges, Faces are there in these Diagrams?

Apologies for the really basic question, however, I don't really understand how to count edges/faces for these types of diagrams with vertices "identified". E.g. For Figure 3.11, Clearly $V=4$, but ...
0
votes
0answers
27 views

Proof that the Klein bottle can be immersed in $\mathbb{R}^3$ and embedded in $\mathbb{R}^4$

We define the Klein bottle as the quotient space of $I^2=[0,1]\times [0,1]$ under the relation $\sim$ for which $(0,y)\sim (1,1-y)$ and $(x,0)\sim (x,1)$. If we found a continous $f:I^2\to R^k$ which ...
1
vote
0answers
21 views

Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
6
votes
0answers
52 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
3
votes
0answers
25 views

How to calculate homotopy invariant winding number?

Consider a map $f:S^1\to U(1)=S^1$, since we know $\pi_1(S^1)=\mathbb{Z}$, which measures how many times the map "wind" around the circle. Given some explicit form of the function $f(\phi$), where ...
5
votes
1answer
45 views

Why is $H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)$?

As the question title suggests, what is the easiest way to see that there is an isomorphism$$H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)?$$
2
votes
0answers
35 views

For any Subspace $A$ of a Path-Connected Space $X$, we have $H_0(X, A)=0$.

I recently learnt about relative homologies and am wondering if the following is true: Statement: Let $X$ be a path-connected topological space and $A$ be a non-empty subspace of $X$. Then $H_0(X, ...
1
vote
1answer
43 views

Trying to understand relative homology group

I'm reading about relative homology group but I'm having hard time in understanding this concept. So I was trying to find $H_1(D^n,S^{n-1})$, but I'm unable to solve this problem. Can someone give ...
1
vote
1answer
45 views

Integral homology of $S^{n-1}/\pi$, $H_*(S^{n-1}/\pi; \mathbb{Z}_p)$

Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\mathbb{C}^n$, $n \ge 2$. ...
0
votes
0answers
29 views

Classifying surfaces

Please help me with part (ii) (i). The polygonal symbol of a certain surface without boundary is $ xy^{-1}x^{-1}zwz^{-1}vyw^{-1}v^{-1}$. Identify the surface. What is the Euler characteristic? ...
2
votes
3answers
124 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
2
votes
1answer
37 views

Is any closed subspace of a $k$-space a $k$-space?

See here for a definition of $k$-space. As the title suggests, is any closed subspace of a $k$-space necessarily a $k$-space?
4
votes
0answers
35 views

Passage to fixed point spaces is object function of a contravariant functor?

Let $X$ be a $G$-space. What is the easiest way to see that that passage to fixed point spaces, $G/H \mapsto X^H$, is the object function of a contravaraint functor $X^{(-)}: \mathscr{O}(G) \to ...
2
votes
2answers
73 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
1
vote
0answers
16 views

Spaces of Labeled Complexes (Munkres)

The following is taken from Munkres' Algebraic Topology book. I tried to determine which spaces (e.g. Mobius Strip, Klein bottle, etc) these complexes are, but to no avail. I computed the Euler ...
1
vote
1answer
59 views

Is every simply connected open subset of $\Bbb R^n$ contractible?

Question: Is every simply connected open subset of $\Bbb R^n$ contractible? I know the result is true for $\Bbb R^2$ because by the Riemann Mapping Theorem every simply-connected proper open ...
3
votes
0answers
38 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. 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 ...
2
votes
1answer
39 views

Why does there exist a deck transformation mapping here?

See Kevin Dong's answer here. Let $\widetilde{\gamma}$ be a path in $X_N$ based at $x_0^N \in X_N$ with $p(\widetilde{\gamma}) = \gamma$ a nontrivial path in $X$. Since the cover is normal, there ...
1
vote
0answers
11 views

Given a triangulation (labeled complex), how do we determine the space?

Given a triangulation, how do we tell which space it is representing? My first idea would be to calculate the Euler Characteristic, but that would still leave some ambiguity, e.g. both the Projective ...
1
vote
1answer
27 views

Intersection preserves homotopy equivalence

Let $Z$ be a topological space with subspaces $X$, $Y$, $X'$ and $Y'$. Suppose that $X$ is homotopy equivalent to $X'$ and $Y$ is homotopy equivalent to $Y'$ do we have that $X\cap Y$ is homotopy ...
1
vote
1answer
30 views

Is this a valid triangulation of a space? (Algebraic Topology)

On the surface, it looks like it would be a Mobius Strip due to the "twist". However, there are some inconsistencies like b is adjacent to d on the left, but not on the right of the figure. ...
1
vote
3answers
49 views

Requirement “closed under finite intersection” in Van-Kampen-Theorem

Given a topological space $X$, a point $x_0 \in X$ and an open cover $\mathcal{U}$ of $X$ of path-connected subsets containing $x_0$ which is closed under finite intersections, we have by Van-Kampen ...
2
votes
1answer
48 views

Counter example to existence of Mayer-Vietoris sequence

Every open cover $X = U \cup V$ gives an exact sequence (called mayer vietoris sequence) $$ \ldots \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \to H_{n-1}(U \cap V) \to \ldots $$ Do $U$ and ...
0
votes
0answers
39 views

Nth Homotopy Group Isomorphic to [T^n, X]

Following Spanier's book on algebraic topology chapter 1, section 6 about suspensions, I'm wondering about the following questions: 1) We know that $S^n$ is an H co-group for all $n\geq1$ because ...
7
votes
1answer
162 views

Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold ...
1
vote
1answer
35 views

What is the meaning of “Continuous Group ”?

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" I did not understand the meaning of this statement " The method is based on finding a nontrivial continuous group ...
15
votes
0answers
114 views

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

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has ...
2
votes
0answers
25 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 ...
3
votes
0answers
61 views

Computational Topology Codes [on hold]

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 ...
1
vote
0answers
50 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 $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
2
votes
0answers
24 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 ...
3
votes
2answers
67 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 $\alpha_0<\dots<\alpha_p$, $$ ...
1
vote
1answer
68 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 ...
16
votes
1answer
278 views
+200

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 ...
2
votes
2answers
71 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 ...
9
votes
1answer
124 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 $X$. ...
5
votes
0answers
52 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
48 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, ...
3
votes
0answers
21 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 ...