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

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4
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67 views

Poincare duality isomorphism problem in the book “characteristic classes”

This is the problem from the book, "characteristic classes" written by J.W. Milnor. [Problem 11-C] Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow ...
4
votes
1answer
60 views

Borel-Moore Homology and Kunneth Formula

Given two algebraic varieties $X$ and $Y$. It is true that $H^{BM}_n(X\times Y) \cong \bigoplus_{i+j=n} H^{BM}_i (X)\otimes_\mathbb{Q} H^{BM}_i(Y)$. I think that the proof is similar to the one in ...
4
votes
1answer
98 views

What does having a basepoint buy us in algebraic topology?

This may be a vague quesion. I am confused between the basepointed case and non-basepointed case in algebraic topology. Is there any convenience in base pointed case? For example, it leads to the ...
4
votes
1answer
179 views

Turn a torus inside out

Let $T=D^2 \times S^1$ be a solid torus, where $D^2$ is a 2-dimensional disk and $S^1$ is a circle. Suppose we have another solid torus $T'$ and we have a homeomorphism $f$ sending a meridian ...
4
votes
1answer
75 views

The groups $[\Sigma^nX,Y]$ versus the homotopy groups

If $X,Y$ are pointed spaces, denote by $[X,Y]$ the pointed homotopy classes of pointed maps $X\to Y$. The sets $[\Sigma^nX,Y]$ actually have the structure of a group for $n\geq 1$. Here $\Sigma$ ...
4
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1answer
100 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
4
votes
1answer
82 views

Reference Request: topological h-cobordism theorem in higher dimensions

The h-cobordism theorem is true in the topological and in the smooth category in dimensions $\ge 6$. (By "dimension, I mean the dimension of the ambient cobordism instead of the dimension of the ...
4
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1answer
33 views

Determining the number of surfaces and boundaries *from* the number of vertices, edges and faces.

Question: Suppose that the number of vertices, the number of edges and the number of faces are given for a set of polyhedra (consisting of triangles only). Can the number of polyhedra and number of ...
4
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1answer
126 views

Homology of the product

I have to prove that $$H_q(X\times\partial I^n,X\times\{p_0\})=H_{q-n}(X)$$ for $X$ a topological space. I tried using induction, but I didn't go too far, and think that using some exact sequence ...
4
votes
1answer
172 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
4
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1answer
54 views

The Hopf invarient with coefficients other than Z.

So generally one defines the Hopf invariant of a map $f: S^{2n-1} \to S^n$ as the coefficient $H(f)$ in $\alpha^2 = H(f) \beta$ where $\langle \alpha \rangle = H^n(C_f)$ and $\langle \beta \rangle = ...
4
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1answer
65 views

Aspherical homology class

I am completely stuck on the following algebraic topology exercise: Let $X$ and $Y$ be CW complexes and $\alpha \in H_p(X)$, $\beta \in H_q(Y)$, $p, q > 0$, homology classes such that the homology ...
4
votes
1answer
249 views

Show that degree of constant map is zero

Let $f \colon S^n \to S^n$ be a constant map, $n > 0$. I want to show that $\deg f = 0$. I will do it by definition. Let $\sum_k g_k \sigma^n_k$ be a singular chain, $g_k \in \mathbb{Z}$, ...
4
votes
1answer
96 views

Gluing axiom of a TQFT

In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT. When they come to prove the gluing axiom, they just mention that "...This statement is ...
4
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1answer
99 views

Finding homotopy equivalence

This is part of a problem from Hatcher: Show that the space in $\mathbb R^2$ which is the union (for $n \in \mathbb N$) of circles $C_n$, where $C_n$ is the circle centered at $(n,0)$ with radius $n$ ...
4
votes
1answer
102 views

Torsion in $H^2(X,\mathbb{Z})$ induced by torsion in $H_1(X,\mathbb{Z})$

Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $$ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow ...
3
votes
1answer
56 views

Induced map on homology by $f\colon S^4 \to S^2 \times S^2$

Show that $$f_* \colon H_4(S^4) \to H_4(S^2 \times S^2)$$ is the zero map for any $f\colon S^4 \to S^2 \times S^2$. We are working with integral coefficients. I tried applying the naturality of ...
3
votes
1answer
62 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
3
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1answer
38 views

Practice Problem Fundamental Group of 7-figured polygon

The question is from Munkres: Consider the space $X$ obtained from a seven-sided polygonal region by means of the labelling scheme $abaaab^{-1}a^{-1}$. Show that the fundamental group of $X$ is the ...
3
votes
1answer
76 views

How do modern categories of spectra manage to avoid “cells now, maps later”?

In Adams' definition, a map $f: X \to Y$ of CW spectra consists of a cofinal subcomplex $X'\subseteq X$ and maps $f_n: X'_n \to Y_n$ that commute with the structure maps. This definition is reproduced ...
3
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1answer
81 views

Cancellation of Direct Product in Top

I'm thinking to the famous problem of cancellation property in Top, i.e: $$T_1 \times T_2 \cong T_1 \times T_3 \Rightarrow T_2 \cong T_3. $$ Clearly there are many counterexamples like $\prod_{i \in ...
3
votes
1answer
37 views

Based covering maps for a bouquet of two circles

For each of the following subgroups of $$ \left \langle x,y \right \rangle = \pi_{1}(S^{1}\vee S^{1}) $$ construct a based covering map $$ \ p:(\tilde{X},\tilde{b})\rightarrow (S^{1}\vee S^{1},b) ...
3
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1answer
46 views

Which p-adic groups are simply-connected?

Suppose that we are working over a nonarchimedean local field $F$, for instance $\mathbb{Q}_p$. Which semisimple algebraic groups (or Lie groups) over $F$ are simply-connected? In particular, I am ...
3
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1answer
68 views

What is the maximal torus in the Lorentz group $O(m,n)$?

I'm close to certain it's just the product of the maximal tori of $O(m)$ and $O(n)$, but I can't quite prove it. I've tried the following: ...
3
votes
1answer
71 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
3
votes
1answer
107 views

Homotopy equivalence -> element in outer automorphism of fundamental group

Let $X$ be a path-connected topological space. for $x \in X, G = \pi_1(X,x)$ Show that a homotopy equivalence $f : X \to X$ gives a well-defi ned element $g \in Out(G)$. How might one begin on this ...
3
votes
1answer
202 views

Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
3
votes
1answer
138 views

Counting roots of polynomial inside $S^1$

I would like to ask for a hint to this problem: Let $p$ a polynomial function on $C$ with no root on $S^1$. Show that the number of roots of $p$ with $|z|<1$ is the degree of the map $q: S^1 \to ...
3
votes
1answer
56 views

Unknotting number formally?

I am reading Colin C. Adams's very nice but not always rigorous "The Knot Book" right now. How does one formalize the unknotting number? (For example, is some restriction on embeddings ...
3
votes
1answer
108 views

How many triangles are there in each “layer” of Poincaré disk?

Assume we grow from a single triangle layer by layer to get the whole disk. Every time a new ring of triangles makes all the vertices of the triangles already in the picture surround by seven ...
3
votes
1answer
113 views

A specific example of a CW complex and a few questions concerning it.

The question I am facing is this one: Construct a CW complex X with a 0-cell x(n) for each natural number $n \geq 0$ and a 1-cell $D_{n}^1, n \geq 1$ which is glued to $x(0)$ at one end and $x(n)$ at ...
3
votes
1answer
104 views

Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
3
votes
1answer
81 views

Misprint in Switzer's Algebraic Topology?

I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can ...
3
votes
1answer
91 views

Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The ...
3
votes
1answer
329 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 ...
3
votes
1answer
207 views

Hawaiian Earring

Let $X=[0,1]$ and $A=\{0\}\cup\{\frac{1}{n}|n\in\mathbb Z\}$. Note that $(X,A)$ is not a good pair. Show that $H_1(X,A)$ is not isomorphic to $H_1(X/A)$. I have a sequence of homology groups: ...
3
votes
1answer
140 views

Basic question about definition of Chern classes

Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)? If we define chern ...
3
votes
1answer
66 views

condition on existence of quillenization

$X_{Ab}$ is the Quillenization of a path-connected space $X$ if $X_{Ab}$ has abelian fundamental group, and there exists a continuous map $X\rightarrow X_{Ab}$ inducing an isomorphism ...
3
votes
1answer
247 views

Sphere with three Möbius strips glued and sphere with a handle and a Möbius strip glued

I am reading the first chapter from Topology by Armstrong. There, after stating the classification theorem for closed surfaces, he has mentioned an example that a sphere with one handle and one Möbius ...
3
votes
1answer
164 views

Covering spaces!

If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it? I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
3
votes
1answer
94 views

What does this free quotient space look like?

Let $S^2=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2=1\}$ and $S^1=\{(s,t)\in \mathbb{R}^2|s^2+t^2=1\}$. Suppose that $\mathbb{Z}/2\mathbb{Z}$ acts on $S^2\times S^1$ in such a way that the generator of ...
3
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1answer
176 views

How does a simplicial map induce a map on chain complexes

I am working on the following exercise: Show that a simplicial map induces a map on chain complexes (hence maps on homology between simplicial complexes). Here is what I have so far... Let $K, L$ be ...
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votes
1answer
30 views

Covering Map of Torus

how can I show that the following map is a covering map of $T:=$ $S^1$ x $S^1$? $\pi: T\rightarrow T$ with $(x,y)$ $\mapsto$ $(x^ay^b, x^cy^d)$, where $a,b,c,d \in \mathbb{Z}$ and $ad-bc=m\neq 0$. ...
2
votes
1answer
29 views

degree of a self map on the sphere

could you help me with this one? How do I determine the degree of the continous map $\mathbb{S}^n\to \mathbb{S}^n$ induced by multiplication with an orthogonal matrix $A$? I think it should be ...
2
votes
1answer
58 views

What is meant by “constant” in Liouville's Theorem?

Liouville's Theorem states that: Every holomorphic* function for which there exists a positive number M such that $|f(z)| \le M$ for all $z \in \mathbb C$ is constant. I'm using this to prove ...
2
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1answer
33 views

Stiefel-Whitney Classes of a submanifold

Suppose we have a manifold $M$ that is the product of $k$ copies of real projective space, say $$ M = \prod_{i=1}^k \mathbb{R} P^{n_i}.$$ Then $H^*(M; \mathbb{Z}/2) = \mathbb{Z}/2 [ \alpha_1, \dots, ...
2
votes
1answer
58 views

Hatcher exercise 2.1.6 (Simplicial homology)

Compute the simplicial homology groups of the $\Delta$-complex obtained from $n+1$ $2$-simplices $\Delta_0^2,...,\Delta_n^2$ by identifying all three edges of $\Delta_0^2$ to a single edge, and for ...
2
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1answer
43 views

What exactly is meant by “an integer basis of the $\mathbb{Z}$-module $H^*(M)$”?

I thought I understood the concept of a cohomology ring, but am confused by the following statement found in a textbook. Context: M is a symplectic manifold of dimension $2n$. "Let us choose an ...
2
votes
1answer
38 views

Showing that identity and g are not homotopic (without Homology)

Question: Are the identity mapping on $S^1$ and the reflection about the $x$-axe homotopic? This is a question which I already know the answer. The objective is to find better answers and suggestions ...
2
votes
1answer
21 views

Why does this open cover of $T^n$ have intersection $T^{n-1}\sqcup T^{n-1}$?

When computing the de Rham cohomology of the $n$-torus $T^n$, usually one takes an open cover $T^n=A\cup B$, where $A=T^{n-1}\times S^1\setminus{N}$ and $B=T^{n-1}\times S^1\setminus\{S\}$, where ...