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

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3
votes
2answers
95 views

Is there some knot theory behind the Mobius donut?

I was watching this video by Numberphile where a professor cuts a bagel into two interlocking pieces. Is this a torus knot or torus link? I'm trying to interpret in terms of $(p,q)$-torus knots Torus ...
2
votes
1answer
74 views

The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
3
votes
1answer
99 views

For a Cantor set $\mathcal{C} \subset S^3$ such that $\pi_1(S^3 \setminus \mathcal{C})=0$, prove $S^3 \setminus \mathcal{C}$ can be split by a sphere.

I'm working from the paper Cantor Sets in $S^3$ with Simply Connected Complements by Richard Skora. On page 184 the second sentence states that any Cantor set $\mathcal{C} \subset S^3$ such that ...
1
vote
0answers
105 views

Gysin sequence and Serre spectral sequence

Given an oriented $S^k$ bundle $E$ over a compact manifold $M$ we get the Gysin Sequence (I am interested in the DeRham cohomology). We can obtain this sequence from the Serre Spectral Sequence if the ...
1
vote
2answers
279 views

CW complex is contractible if union of contractible subcomplexes with contractible intersection

Exercise 0.23 from Algebraic Topology by Hatcher reads: Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible. I have ...
4
votes
0answers
163 views

Higher homotopy groups: Basepoint independence.

Let $f,g: [0,1]^n=I^n \rightarrow X$ be contiuous maps s.t. $f(\partial I^n)=g(\partial I^n)=x_1$. If $\gamma:I \rightarrow X$ is a path joining $x_0$ and $x_1$ ...
2
votes
0answers
76 views

Classifying covering spaces of product spaces

Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we ...
1
vote
0answers
37 views

Geometric definition of the stable commutator length

In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the ...
0
votes
1answer
84 views

Fundamental group of disjoint union of two 2-tori identifying them along pairs of points

I'm trying to solve the following problem: let $X$ be the space obtained from the disjoint union of two 2-tori $A,B$ be identifying them along 2 pairs points (resp. three pairs of points). If we ...
2
votes
1answer
108 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
0
votes
2answers
61 views

Induced map on fundamental groups between surfaces

Let $\Sigma_n$ and $\Sigma_m$ be two closed oriented surfaces of genus $n$ and $m$, with $n \leq m$. We may think about these surfaces as connected sums of tori, so there is an canoical inclusion map ...
2
votes
1answer
91 views

Cell structure of $S^2 \times S^1$

Can anyone please provide the cell structure of $S^2 \times S^1$? I know that there are one cell in each dimension from 0 to 3 but I am not sure about the attaching maps. Thanks in advance.
1
vote
1answer
49 views

cohomology is dual to homology of a spectrum if homology is free

Let $E$ be a multiplicative spectrum (and $X$ a space with $H_n(X; \mathbb{Z})$ free abelian for every $n$). The following excerpt is taken from the notes here claim that item (1) below easily implies ...
0
votes
1answer
76 views

Condition for Orientability of Manifold

Let $M^n$, $n>2$ be a manifold and let $f:D\rightarrow M$ be an embedding of the closed $n-$disk in $M$. Prove or Disprove: $M$ orientable iff $M-f(D)$ is orientable. $M$ is orientable iff all ...
2
votes
1answer
53 views

Quotient of Unit Quaternions by Subgoup (Lie Groups)

Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
3
votes
1answer
62 views

Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$

i.e. Does $\mathbb{R}P^n$ have a tubular neighborhood $N$ such that $N-\mathbb{R}P^n$ is disconnected. My guess is yes, but don't know how to show it convincingly ( or maybe only for $n$ odd, I'm ...
1
vote
1answer
76 views

Fundamental group computation

I am trying to compute the homology groups and the fundamental group of the space $X$ obtained as the disjoint union of a circle and a cylinder $S^1\times I$ by attaching the cylinder along its ...
4
votes
1answer
67 views

True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
6
votes
1answer
222 views

An algebraic topology proof of a result from analysis

A colleague of mine recently brought up the following result from real analysis: Theorem: If $f:\mathbb{R}^2\to\mathbb{R}^2$ is continuous and $|f(x)-f(y)|\geq |x-y|$ for all $x,y$, then $f$ is onto. ...
0
votes
1answer
42 views

Proof for Homologous cycles

Prove that two cycles that surround the same holes differ by a boundary i.e. the relation for calling two cycles homologous as mentioned here. ...
1
vote
0answers
43 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
2
votes
3answers
576 views

Holomorphic functions on a complex compact manifold are only constants

Is there a simple proof that every holomorphic function $M\to\mathbb{C}$ on a compact complex manifold $M$ is constant?
0
votes
1answer
71 views

Exercise 2, chapter 4, Hatcher.

Show that if $\varphi: X \rightarrow Y$ is a homotopy equivalence, then the induced homomorphisms $\varphi_{*}:\pi_n(X,x_0) \rightarrow\pi_n(Y,\varphi(x_0))$ are isomorphisms, for all n$\in ...
2
votes
1answer
117 views

Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the examples ...
3
votes
1answer
127 views

Characteristic Class with arbitrary coefficient

I'm looking for some "natural definition" for characteristic class with arbitrary coefficient. For example, the chern class satisfies four conditions and Stiefel-Whitney class satisfies three ...
0
votes
0answers
73 views

Intuition behind certain examples of fundamental groups

I have some intuition behind the interpretation of having nontrivial fundamental group, detecting the holes in the space and so on. But I don't quite see how interpret the fact that the fundamental ...
4
votes
1answer
61 views

definitions of various spectra: $E^X$ and $E \wedge \Sigma^\infty X$

Let $E=\{E_n\}$ be a spectrum given by a sequence of pointed CW complexes $E_n$ and inclusions $\Sigma E_n \to E_{n+1}$. Let $X$ a pointed CW complex. I had a few very naive questions I had while ...
3
votes
0answers
84 views

Motivation behind definition of homologous cycles

Two cycles are said to be homologous if their difference is a boundary.(usual meanings implied) What is the motivation behind this definition or the intuitive meaning it carries. I am looking of ...
5
votes
2answers
151 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
-1
votes
1answer
63 views

Powers of Orbifold Fundamental Groups

I have reduced a problem to $\pi(Y)^n/G^n$ where Y is a manifold and G is a group acting on the manifold. Can I "factor out," the $n$? i.e. $(\pi(Y)/G)^n$. Note that $\pi(X)$ is the fundamental group ...
4
votes
1answer
122 views

Question about the Betti numbers

Definition of Betti number at http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the maximum amount ...
4
votes
1answer
92 views

Simplicial homology of the skeleton of a simplex

Let $n$ and $k$ two natural numbers. We consider the (abstract) simplicial complex $K$ on $n$ vertices $v_1,\dots,v_n$ and such that a subset of $\{v_1,\dots,v_n\}$ is a face of $K$ if and only if it ...
3
votes
1answer
115 views

simple closed curve is nullhomologous iff is separable

A simple closed curve $\gamma$ in an orientable genus $g$ surface $M$ is nullhomologous if and only if $M \setminus \gamma$ consists of two connected components, one of which is a surface $N$ with ...
1
vote
0answers
82 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
0
votes
1answer
61 views

Singular Chain of a Hyperplane.

I refer to the definitions of Hatcher's Algebraic Topology. Is it possible to model a hyperplane $H$ (or half of it) of $\mathbb{R}^n$ with a singular chain? And if - how would its boundary look like? ...
0
votes
1answer
91 views

Reduced suspension and unreduced suspension

In May's "A concise course in Algebraic Topology" Chap 14 section 1, the author says $\Sigma (X_+)$ is $\Sigma X\vee S^1$ where $X$ is an unbased space and $X_+$ is the union of a disjoint basepoint ...
2
votes
1answer
139 views

“Equivalent” definitions of the gluing axioms

I tried to convince myself that the two caracterizations of a presheaf that is a sheaf given in wikipedia are equivalent but I couldn't. (F presheaf and notations from wiki) Let's take a simple ...
1
vote
1answer
81 views

Show that a star convex set $X \subset \mathbb{R^n}$ is simply connected.

Show that a star convex set $X \subset \mathbb{R^n}$ is simply connected. I have the idea of the proof : There is a natural retraction of $X$ onto the set $\{x_0\}$, because every other point $x \in ...
4
votes
1answer
92 views

Fundamental group of quotient of $S^1 \times [0,1]$

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$. I've been trying to ...
1
vote
2answers
169 views

A doubt in Hatcher's Algebraic Topology.

I refer to pg. 27 of Hatcher's Algebraic Topology. I refer to the part where Hatcher proves that $f.(g.h)\cong (f.g).h$ For the life of me, I cannot figure out how the diagram on the right proves ...
0
votes
1answer
70 views

Why does two joined circles (wedge sum) with a point removed deformation retract to a single circle?

Let $X$ and $Y$ be two unit circles in $\mathbb{R}^2$. $X$ is centered at $(1,0)$, while $Y$ is centered at $(-1,0)$. Consider $A = X \cup Y$. Let $p \in Y$ be a point that is not $(0,0)$. Why is ...
0
votes
1answer
121 views

Are there big implications of Poincare conjecture?

I was just curious: are there any big corollaries of Poincare conjecture in dimension $3$? Is it useful to prove some other (big) theorems? Or is it just a nice statement, and its main value is that ...
1
vote
1answer
97 views

Question about $\pi_0(X)$

I'm studying the fundamental group of a topological space and I've studied a proof checking that $\pi_1(X)$ is a group. I'm thinking if the space of path components $\pi_0(X)$ is a group or not. ...
3
votes
1answer
98 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
1
vote
1answer
76 views

Definition of boundary in a topological invariant way

I'm reading through Aguilar & Prieto lecture notes "Fiber bundles" (available online by googling it, ...
4
votes
2answers
118 views

Null-homotopic covering space map

I'm stuck with the following question, which looks quite innocent. I'd like to show that if a covering space map $f:\tilde{X}\to X$ between cell complexes is null-homotopic, then the covering space ...
4
votes
1answer
58 views

Homotopy classes of maps from the projective plane to $S^1 \times S^3$

I have a past qual question here: characterize the space $[(\mathbb{RP}^2,x),(S^1 \times S^3,y)]$ of homotopy classes of maps from $(\mathbb{RP}^2,x)$ to $(S^1 \times S^3,y)$, where here $x \in ...
2
votes
2answers
91 views

intersection number of twocompact oriented manifolds

I have an oriented manifold M of n dimension and 2 oriented submanifolds, one of dimension k and the other of dimension n-k , I have to understand which is the intersection number of those manifolds. ...
0
votes
1answer
106 views

Homology of 3-sphere minus an embedding of $S^1 \times \mathbb{D}^2$

I'm having trouble with the following past qual question: Let $\phi \colon S^1 \times \mathbb{D}^2 \hookrightarrow S^3$ be an embedding, where $\mathbb{D}^2$ is the open unit disk in $\mathbb{R}^2$. ...
4
votes
1answer
65 views

Homotopy groups relating to toric varieties

It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, ...