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

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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 ...
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0answers
20 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 ...
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1answer
21 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 ...
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1answer
35 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 ...
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2answers
45 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 ...
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1answer
40 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.
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1answer
27 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 ...
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1answer
28 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 ...
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1answer
32 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 $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
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1answer
36 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 ...
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1answer
44 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 ...
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26 views

Non-normal covering space of a Klein bottle

I'm trying to construct a non-normal covering space of the Klein bottle by a torus. An old question addressed this issue and an answer to it produced a non-normal subgroup of $\pi_1$ out of the blue. ...
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1answer
43 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 ...
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1answer
66 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. ...
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1answer
29 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. ...
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0answers
32 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 ...
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3answers
235 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?
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1answer
41 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 ...
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1answer
79 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 ...
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0answers
23 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 Steifel-Whitney class satisfies three ...
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0answers
52 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 ...
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0answers
37 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 ...
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0answers
43 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 ...
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2answers
76 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) ...
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1answer
48 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 ...
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1answer
44 views

Question about the Betti numbers

can someone explain me this definition from :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 ...
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1answer
41 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
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1answer
63 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 ...
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0answers
65 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 ...
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1answer
52 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? ...
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1answer
41 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 ...
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2answers
98 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 ...
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1answer
35 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 ...
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1answer
68 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 ...
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2answers
138 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 ...
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1answer
39 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 ...
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1answer
79 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 ...
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1answer
75 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. ...
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1answer
48 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 ...
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1answer
45 views

Definition of boundary in a topological invariant way

I'm reading through Aguilar & Prieto lecture notes "Fiber bundles" (available online by googling it, ...
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2answers
75 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
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1answer
43 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 ...
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2answers
30 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. ...
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1answer
65 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 $\mathbb{R}^2$. ...
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1answer
43 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, ...
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0answers
38 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
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0answers
67 views

Why Must the Degree of this Map be 0? [closed]

Let $f:S^3 \rightarrow S^1\times S^1\times S^1$ be a continuous map. Show that it's degree must be $0$. (Just a hint would be good)
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2answers
77 views

Cohomology Ring of Klein Bottle over $\mathbb{Z}_2$

I am trying to show that the cohomology ring of the Klein bottle with $\mathbb{Z}_2$ coefficients is $H^*(K,\mathbb{Z}_2) \cong \mathbb{Z}_2[x,y]/(x^3,y^2, x^2y)$. What I know: ...
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1answer
22 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
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1answer
29 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...