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

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-3
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1answer
115 views

An application of Euler Characteristic to Tetrahedron Packing

The following is an application of Euler's equation to tetrahedron packing of any convex polyhedron. I related it to Euler formula; consequently, a third equation is obtained which is independent of ...
2
votes
1answer
51 views

Exact sequences of bundles and orientations

If we have an exact sequence of finite-dimensional vector spaces $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow0$$ then an orientation of any two induces an orientation of the third. I have ...
2
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1answer
71 views

The identification $G=\Omega^\infty \Sigma^\infty S_0$ of the stable group of self homotopy equivalences of spheres with the suspension spectrum

My topology professor told me in a discussion that the suspension spectrum $colim \Omega_n \Sigma_n S_0$ is the same as the monoid $G$ where $G=colim G_n$ where $G_n$ are self homotopy equivalences of ...
2
votes
0answers
46 views

Difference between product projections and split epis in $\mathbf{Top}$

I don't understand the excerpt below. The trivial bundles were defined as split epimorphisms, and since the ground category was additive with kernels (in fact abelian), they are the same as ...
0
votes
1answer
56 views

Relative homotopy

Show, that the functions $g: S^1\to S^2$, $(x,y)\mapsto (x,y,0)$ and $h: S^1\to S^2$, $(x,y)\mapsto (x,-y, 0)$ are relative homotopies to $(1,0)\in S^1$ Hello, I have a question to this task. I ...
3
votes
0answers
49 views

Isomorphism in integral Cohomology gives isomorphism in rational cohomology

I was asking myself the question, if a map $f\colon X \to Y$ between CW complexes gives an isomorphism between $H^*(X,\mathbb{Z})$ and $H^*(Y,\mathbb{Z})$ does it already give an isomorphism between ...
1
vote
2answers
51 views

Compact cohomology group of connected n-dimensional connected oriented manifold

I know how to show $H_c^n(M)\simeq\mathbb{R}$, where M is a oriented connected n-dimensional manifold, by showing the integration map is isomorphism. However, I found in the book that this is a ...
0
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0answers
16 views

Structure group reduction criterion in terms of classifying map lifting

I am looking for a proof of the following theorem which is given as an exercise in Ralph Cohen's notes on the topology of fiber bundles pp.74. But in view of its importance to the later chapters I ...
2
votes
1answer
52 views

Homeomorphism between N-disk and N-Projective Plane

I've just showed that: $D^n$, quotiented with this equivalence relation: $x\sim y \iff x=-y \text{ and } x,y\in\partial D^n$ (i.e. the antipodal points on the boundary of $D^n$ are identified) is ...
0
votes
2answers
31 views

Inverse mapping of a contractible space

So, let us be given a nonconstant continuous function $f:X \to Y$ and $B\subset Y$ is contractible. Now, I wonder if $f^{-1}(B)$ is contractible. I need this to solve a problem correctly(if you can ...
1
vote
1answer
50 views

$SO(3)$ homeomorphic to $\mathbb{R}P^3$

I'm doing some topological base-exercises, but I can't come up with this problem (That I suppose should be quite trivial): $SO(3)$ is homeomorphic to $\mathbb{R}P^3$. Any hints? thank you in Advance!...
3
votes
1answer
38 views

Proof of :$H^0(E;\pi_0E)\cong \hom_{\pi_0}(H_0(E;E_0);\pi_0E)$ for $E$ a multiplicative spectrum.

Let $E$ be a multiplicative spectrum, connective, and assume $\pi_0E$ is cyclic. I want to prove that $$H^0(E;\pi_0E)\cong \hom_{\pi_0E}(H_0(E;\pi_0E);\pi_0E)$$ Recall that $H^0(E;\pi_0E):= [E; K(\...
0
votes
1answer
54 views

Coefficients of homology

I am wondering why people use different coefficients when defining homology of simplicial complex, like homology over $R$, $Z$, $Z/2$, etc? Is one better then the other and why? Moreover, which one(s)...
3
votes
0answers
22 views

Complexes $K$, $L$, imply $|K| \cap |L|$ is polyhedron.

I am using Armstrong's topology text, and have been really stumped on what this question is asking. It says If $K$ and $L$ are complexes in $\mathbb{E}^n$, show that $|K| \cap |L|$ is a ...
0
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0answers
9 views

Iff conditions for acyclic, free, positive chain complexes with augmentation

I have some doubts about the formulation of the following lemma (from Ferrario, Piccinini - Simplicial structures in topology) and its proof. (II.3.8, page 72) Lemma. Let $(C,\partial) $ be a ...
0
votes
1answer
26 views

Why is the quotient map $G\rightarrow G/T$ a fibration?

I have just learnt about fibrations and I saw somewhere the following. Given a compact lie group $G$, one can consider a maximal torus $T$ of $G$ then the quotient map $G\rightarrow G/T$ is a ...
2
votes
0answers
18 views

Pairing on the AHSS induced by cap product: why does it exists

This is my setting: Let $\xi \colon X \to B$ be a vector bundle. Let $E$ be a ring spectrum. Suppose given a natural cap product $$ \frown \colon E^s(X,X\setminus B)\otimes E_{s+t}(X,X\setminus B)...
0
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0answers
41 views

Weibel 5.1.1 Exercise

I know this topic is already dealt on Total complex homology exact sequence, But have a question on the answer. The answer says that $$H_{p + q}(T) \cong \frac{\{(a,b) | d^v_{p-1,q+1}(a)+d^h_{p,q}...
3
votes
0answers
31 views

Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
0
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2answers
83 views

Reference for Algebraic Topology

I know undergraduate algebra (groups, rings, fields, Galois etc), undergraduate differential geometry, undergraduate real/complex analysis and now I feel as though to get to the next level, i should ...
0
votes
1answer
42 views

What is the topological degree of the constant map?

What is the topological degree of the constant map? To me it does not make any sense, once $f$ being the constant map has no regular values. So, how to proceed?
3
votes
2answers
59 views

Why $h$ has zero topological degree?

I am trying to prove that $f,g : M^n \to S^n$, both $C^1$ (indeed just $C^0$ is enough) with the same topological degree are homotopic. I saw on a book that the trick is as follows: Take $W = M\...
1
vote
1answer
51 views

What is the classifying space G/Top?

I simply can't find the definition(except in one book on surgery where a definition was not actually given but instead they alluded to what the definition is) and I have spent an hour and half looking....
1
vote
1answer
50 views

Subgroup of Finite Index Containing a Given Finitely Generated Subgroup of a Free Group: Problem 12 in 1A in Hatcher.

Problem 1A.12 (Hatcher) Let $F$ be a finitely generated free group and $H$ be a finitely generated subgroup of $F$. Let $x\in F-H$. Show that there is a finite index subgroup $K$ of $F$ such that $H\...
1
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0answers
39 views

Homotopy, topology

Let $X$ be a topological space. Let $w$ be path in $X$. $\overline{w}(t)=w(1-t)$ and $\iota_{x}(t)=x$ for every $t\in[0,1]$ and $x\in X$. Give homotopies $H$ and $K$, with $w\ast\iota_{...
0
votes
1answer
29 views

Covering map associated with open cover

Let $ \left\{U_i \right\}$ be an open cover of $X$. On some online sources and some MSE questions, the map $\coprod _iU_i\rightarrow X$ is given as an example for a local homeomorphism which is not a ...
3
votes
1answer
44 views

How is it possible that $H_p(\Bbb{S}^n)\cong H_p(\Bbb{S}^{n-1})$?

My Algebraic Topology book says the following: For $\Bbb{S}^n$, $H_p(\Bbb{S}^n)=\Bbb{Z}$ for $p=\{0,n\}$, and $H_p(\Bbb{S}^n)=0$ otherwise. Also, by Mayer-Vietoris, $H_p(\Bbb{S}^n)\cong H_p(\...
5
votes
1answer
57 views

Explicit verification of signs in Morse complex

I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to $\partial^2=0$. The books I've looked in seem to say either ...
0
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0answers
24 views

Book with Chapter on Fundamental Polygons

Does anyone know of a book with a chapter explaining fundamental polygons? By Fundamental Polygon I mean for example, Fundamental polygon of Klein Bottle, as shown below. I understand it is a ...
1
vote
1answer
34 views

Hatcher question: How to Cut and Glue from Tetrahedron to Klein Bottle

I have been stuck on this question for a long time: Show that the $\Delta$-complex obtained from $\Delta^3$ by performing the edge identifications $[v_0,v_1]\sim [v_1,v_3]$ and $[v_0,v_2]\sim [v_2,...
0
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0answers
22 views

Munkres algebraic topology section 25 question 6 (Mayer Vietoris)

This question is from Munkres Algebraic Topology section 25 question 6 The question, vertabim, says. "We shall study the homology of $X\times Y$ in chapter 7.For the present, prove the following, ...
0
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0answers
15 views

Representing Covering Spaces by Permutations: Proof Verification.

$\newcommand{\FG}{\pi_1}$ Given a covering projection $p:\tilde X\to X$, and $x_0\in X$, we can naturally define a \emph{right} action on $F=p^{-1}(x_0)$. For each point $\tilde x\in F$, and each $[\...
4
votes
2answers
54 views

What is the relationship between diffeomorphisms of the sphere modulo isotopy and exotic spheres?

In his "Classification of (n-1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", Milnor observes that since his exotic 7-spheres admit a Morse function with only two critical ...
0
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0answers
21 views

Compute $h^{n+k}(D^n\times \mathbb{R}^k, D^n\times (\mathbb{R}^k\setminus \{0\}))$. Strange result

Let $h^*$ be a generalised cohomology induced by a multiplicative spectrum $E$. The book I'm following, (Kochman) make the following claim $$h^{n+k}(D^n\times \mathbb{R}^k, D^n\times (\mathbb{R}^k\...
4
votes
1answer
64 views

Is this relative homology equals to wedge sum of two tori?

If $X$ is a connected sum of tow tori, and $A$ is its center circle as shown in picture below. I would like to compute $H_n(X,A)$. There is a statement in the book that $H_n(X,A)$ represents the ...
5
votes
1answer
41 views

Smallest number of $n$-simplices in a triangulation of the sphere

Let $X$ be a simplicial complex homeomorphic to $S^n$. I proved that there must be at least $(n+2)$ vertices in $X$ and that there must be at least one $n$-simplex in $X$. Now I want to prove that ...
5
votes
1answer
115 views

Poincare lemma for compact vertical supports in Bott & Tu

I'm trying to work out Proposition (6.16) from Bott and Tu which states that $H^*_{cv}(M\times\mathbb{R}^n)$ is isomorphic to $H^{*-n}(M)$. The first group being forms compactly supported along the ...
3
votes
0answers
14 views

Versal deformation of $x^3+y^3$

I am trying to compute fundamental group of complement to discriminant hypersurface of $f=x^3+y^3$ singularity via Zarisski-van Kampen theorem. So, I need a versal deformation of singularity to ...
3
votes
1answer
69 views

Definition of $\pi_0$

$\pi_0(X)$, for a topological space $X$, is the space of homotopy classes of maps $S^0\to X$. I suppose here $S^0$ may be taken as the set $\{\pm1\}$ with the discrete topology. I am wondering, are ...
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0answers
151 views

Why does excision imply this?

In exercise $4$, page 230 of Bredon, he asks for a proof of the Mayer-Vietoris sequence using a commutative braid diagram which substitutes some terms by others using excision. I've solved the ...
0
votes
1answer
22 views

Principal bundle built from equivariant CW complex

The Milnor join construction of classifying space for $G$ is a $G$-equivariant CW complex $\mathcal{J}(G) = \lim_{k\to \infty}{G^{*(k+1)}}$ which admits a free $G$-action. Then the projection $p: \...
3
votes
0answers
29 views

$E$-orientation of a closed manifold induce $E$-orientation of normal bundle: passage in the proof.

I'm trying to follow Kochman's proof of the well-known result For a ring spectrum $E$, and closed manifold $M^n$ together with an embedding in $\mathbb{R}^{n+k}$, the following are equivalent: ...
2
votes
1answer
50 views

CW complex structure of geometric realization

In Ralph Cohen's notes on the topology of fiber bundles he makes the following claims: on pp.69, he says the geometric realization of a simplicial set is a CW complex on pp.70, he says the geometric ...
2
votes
0answers
32 views

Homotopy type of mapping space

In Ralph Cohen's notes on the topology of fiber bundles (pp.63) he claims that the space of all $G$-equivariant maps from $P$ to $EG$ denoted by Map$^G(P,EG)$ is aspherical, where $EG$ is the total ...
4
votes
1answer
71 views

Can $\mathbb{Z}/6\mathbb{Z}$ act freely and properly discontinuously on $\Sigma_4$?

Let $\Sigma_m$ denote the closed connected orientable surface of genus $m$. Let $N_m$ denote the closed connected non-orientable surface of genus $m$. I was wondering which cyclic groups could act ...
3
votes
2answers
63 views

Take tautological bundle over $\mathbb{C}P^n$ less the zero section. Is this homeomorphic to $\mathbb{C}^{n+1}-{0}$?

It appears that (non zero vector in $\mathbb{C}^{n+1}$)$\mapsto$(line containing the vector, vector in this line) is a homeomorphism from $\mathbb{C}^{n+1}$ to the tautological bundle over $\mathbb{C}...
1
vote
1answer
18 views

Proving that for a multiplicative (B,f)-structure $\mathfrak{B}$ (or X-strcture, or B-structure), Thom spectrum $M\mathfrak{B}$is a ring spectrum.

I'm interested in filling the detail of the claim I made above. I'm following Kochman's notation (page 14 for a def.). Actually he never claims it, (he never spoke about ring spectra), but I think ...
0
votes
1answer
42 views

Vector field on n-manifold whose sum of indexes is equal to Euler charasteristic

For 2-manifolds and 3-manifolds such a tangent field (whose singular points indexes sum to manifold's Euler chracteristic) construction can be done visually. For example, for triangulated 2-manifold ...
2
votes
1answer
45 views

Why does $f(x,z)=(x,z^2/|z|)$ have degree $2$?

Write the $n$-sphere as the set $S^n\approx \{(x,z)\in\Bbb R^{n-1}\times\Bbb C: |x|^2+|z|^2=1\}$, and define a mapping $f: S^n\to S^n$ by $f((x,z)) = (x,\frac{z^2}{|z|})$. Why is $\deg(f)=2$? ...
2
votes
1answer
51 views

All cohomology of quadrics comes from algebraic cycles

I've read in a number of place now the statement that all cohomology of quadrics (complex projective ones) comes from algebraic cycles, but I cannot find any source for this. So I hope someone here ...