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

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fundamental theorem of algebra proof

I have seen some proofs of the fundamental theorem of algebra using algebraic topology. But I have seen an exercise to prove the theorem by defining this map $f_t: S^1\longrightarrow S^1$ by $f_t(z) ...
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88 views

Extension of Brouwer's degree to continuous functions.

I am studying the first chapter of this book: Topological Degree Theory and Applications At page 13 of this document, Definition 1.2.5, it is essentially said that to define the degree of a ...
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80 views

k3 surface as ramified double cover of $\mathbb{P}^2$

I read that one example of k3 surface is a double cover of $\mathbb{P}^2\mathbb{C}$ ramified over a sextic. My question is why a sextic? i believe that the sextic is isomorphic to the ramification ...
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114 views

Cech Cohomology Of Pullback Linebundle

my question is as follows. Let $\chi$ a compact Calabi-Yau 3-fold and $A,B \subset \chi$ two 2-complex dimensional manifolds such that their intersection $C := A \cap B$ is a 1-complex dimensional ...
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43 views

isomorphisms of $\pi_{1}(T^2, x_{0})$ with itself.

I1m studying fundamental group and its relation with covering maps, I was thinking about an exercise: every isomorphism of $\pi_{1}(T^2, x_{0})$ with itself is induced by a homomorphism $f:T^2 ...
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39 views

does a method exist to distinguish two component link consisting of just two unknots from an unlink?

Clearly, linking number is not enough as there are links like whitehead. There is the enhanced linking number based on conway polynomial that can distinguish whitehead (and infinite family of such ...
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73 views

A detail in the proof of Poincaré duality

In Hatcher's Algebraic Topology, on page 246 (here, in the book), about two-thirds down the page he states that showing the commutativity of the two squares shown in the diagram, not involving the ...
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102 views

The fundamental group of the union of three convex open subsets of $ \mathbb{R}^n$.

I have to prove that the fundamental group of the union of three open convex subsets of $\mathbb{R}^n$ is trivial or $\mathbb{Z}$. I can show that it has only one generator, but I can't prove that if ...
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81 views

Construct a space with free involution and homological restriction

I'm looking for a space $X$ which satisfies the following conditions: $X$ is a compact manifold. $H_\ast (X;\mathbb Z)$, the integral homology groups $X$, are torsion free. There is a free ...
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81 views

Cohomology Finite covering and Weyl group

Let $G$ a compact Lie group, $T$ a maximal torus in $G$ and $W=N(T)/T$ its Weyl group. Then we have a finite covering (why is a covering?) $ W \rightarrow G/T \rightarrow G/N(T) $ Has $G/N(T)$ a ...
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139 views

Torsion-free fundamental group.

Is there a name for spaces whose fundamental group has no torsion? And what, if any, are some nice properties of these spaces?
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73 views

A proof of simply connectedness of a symplectic quotient

Let $\rho$ be a unitary representation of a torus $G$ on $\mathbb{C}^n$. The action of $\rho$ is Hamiltonian with a moment map $\mu:\mathbb{C}^n \to \mathfrak{g}^*$. Here $\mathfrak{g}^*$ is the dual ...
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402 views

Fundamental group of an orientable surface of infinite genus.

I am trying to calculate the fundamental group of an orientable surface $X$ of countably infinite genus. The $1$-skeleton $Y$ of $X$ is infinite wedge of circles, so its fundamental group is free ...
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58 views

Compacity in the homological definition of orientation.

For a manifold $M^n$, orientation is often defined as a globally consistent choice of local orientations ie. a choice of generators $\mu_x$ of $H_n(M,M-x;R)$ (this group is isomorphic to R by ...
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94 views

Products of CW-complexes

I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using ...
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124 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 ...
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168 views

Construction of a sphere bundle

Let $\pi:E\to M$ be a rank $k$ vector bundle over a compact manifold $M$. The usual method to associate a sphere bundle to $E$ is by considering only vectors of length 1 in each fiber of $E$ (after ...
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78 views

Showing $\Omega^n(X,x_0)\approx M(S^n,*; X,x_0)$

I am trying to prove $$\Omega^n(X,x_0)\approx M(S^n,*; X,x_0)$$ where $\Omega^n(X,x_0)$ is the $n$-loop group and $M(S^n,*; X,x_0)$ is the set of pointed continuous maps from $(S^n,*)$ to $(X,x_0)$ ...
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153 views

Fundamental Group!

Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space ...
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121 views

cohomology isomorphism

Let $X$ be a finite dimensional CW complex and $A$ be a closed subset in $X$ and $N$ a regular neighborhood of $A$ that deformation retracts onto it. why do we have for each $i$, $$H^{i}(X-A;\mathbb ...
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53 views

Unramified functions between Riemann surfaces

Let $F:X\rightarrow Y$ a unramified holomorphic function between two compact Riemann surfaces. I don't understand why $F$ is a covering map. By a well-known theorem $F$ is surjective; then since the ...
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329 views

Fibre and homotopy fibre

$p:E \rightarrow B$ is a fibration and $F$ is its fibre and $F_p$ its homotopy fibre. If $i:F \rightarrow F_p$ is the inclusion, is there a homotopy inverse $r$ of $i$ such that $r \circ i = id$?
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60 views

Chain map variant

A map $\phi_*:C_*\rightarrow D_*$ of chain complexes is a chain map if it intertwines the boundary operator, i.e. $\phi\partial=\partial \phi$. It is well known that such a map descents to homology, ...
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360 views

Universal cover of wedge product of circles

I want to ask a question about universal covering of wedge space of two circles. It is known that the universal covering space is the cayley graph. I have another thing in mind which I came up with ...
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67 views

For which $g,p$ does $\Sigma_{g,p}$ cover $\Sigma_{3,2}$?

I am preparing for my qualifying exams. There is an algebraic topology problem I don't know how to do it. Thanks a lot for your help. Let $\Sigma_{g,p}$ denote the surface of genus $g$ with $p$ ...
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136 views

Why is well-pointedness necessary for $X\hookrightarrow M_f$ to be a pointed cofibration?

In Jeffrey Strom's Modern Classical Homotopy Theory on page $125$, it is stated that "Now we come to one of the crucial differences between the pointed and the unpointed categories. The mapping ...
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207 views

Definition of a topological module

A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
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101 views

A question about the proof $\pi_1(S^1,1) \cong \mathbb{Z}$

I am working through the proof that the fundamental group of $S^1$ is isomorphic to $\mathbb{Z}$ from the book Basic Topology by Armstrong. There they are defining a map $\pi: \mathbb{R} \to S^1$ by ...
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106 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
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128 views

tensor product of two chain homotopic maps are again chain homotopic?

Let $C$,$C'$, $D$, $D'$ be chain complexes, $f$, $f'\colon C\to C'$ and $g$, $g'\colon D \to D'$ two pairs of homotopic chain maps.How to show $f \otimes g$ and $f' \otimes g' \colon C\otimes D\to ...
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225 views

Double Coverings of the Double Torus

I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is $$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$ where ...
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331 views

Covering space calculation of figure eight.

I'm trying to do this calculation in Hatcher. So for the (1). I imagine cutting the loop at $a$ on the left call this Y and cutting the loop $a$ on the right call this Z. This will give you two ...
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920 views

Torus as double cover of the Klein bottle

Reading through some lecture notes and it says The torus $T^2$ is the orientation double cover of the Klein bottle $K$, via the covering projection $p:T^2\to K; [x,y]\mapsto [x,2y]$ Could someone ...
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103 views

Why is $\pi_2(S^1 \vee S^2)$ not finitely generated?

This is an exercise following a discussion of fibrations. preceding that there was a discussion of cofibrations and the long exact sequence of homotopy groups of a pair. Any hints would be greatly ...
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224 views

Does homotopy equivalence of pairs $f:(X,A)\to(Y,B)$ induce the homotopy equivalence of pairs $f:(X,\bar A)\to(Y,\bar B)$?

When we have a homotopy equivalence through a pair $f:(X,A)\to (Y, B) $, it is said that we can induce a homotopy equivalence through a pair $f:(X,\bar A)\to (Y,\bar B) $, where $\bar A$ stands for ...
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240 views

Covering a connected sum

I have the following problem as a part of my homework: Let $S$ be a closed surface (compact and connected). Show that for every $k$ exists a covering map of $k$ folds $p_k:S_k \rightarrow ...
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262 views

question about the short exact sequence arising from a fibration

Let $F\hookrightarrow Y\stackrel{f}{\longrightarrow} B$ be a fibration. If $F$ is contractible in $Y$ via some homotopy $H: F\times I\rightarrow Y$, we get split short exact sequences: $0\rightarrow ...
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167 views

Simplicial complexes and deformation retracts

I spent a couple of hours today trying to prove the following: Let $L$ be a subcomplex of a simplicial complex $K$. Let $U_L$ be the union of the relative interiors of the relative interiors of all ...
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165 views

Deformation retracts of CW complexes

I'm tearing my hair out trying to prove that a contractible subcomplex ,$K$, of a contractible CW complex, $L$, is a strong deformation retract of $L$.   What I have so far: I can show that the ...
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405 views

Hatcher 1.3. problem 16

Given maps $X\to Y\to Z$ such that both $Y\to Z$ and the composition $X\to Z$ are covering spaces, show that $X\to Y$ is a covering space if $Z$ is locally path-connected.
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56 views

An unexplained iso between $H^{m+1}(O(m+2)/O(m), S^m)$ and $H^{m+1}(S^{m+1})$

I am reading topology of Lie groups by Mimura and Toda and got to the part where they are beginning to compute $H^*(O(n))$, page 120. If we let $r_m :S^m \to O(m+1)$ be the map that sends $v$ to the ...
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475 views

Covering space of the wedge of the unit circle and the real projective plane

Let $Z * Z/2Z = \langle a, b | b^2=1\rangle$ be represented by $X = S^1\vee RP^2$ i.e. the wedge of the unit circle and the real projective plane. Let $H$ be the smallest normal subgroup containing ...
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67 views

Are PL maps determined by PL-paths?

Let $X,Y$ be polyhedra. If $f\colon X\rightarrow Y$ is a PL map and $g\colon I\rightarrow X$ is a PL map (where $I$ denotes the interval), then $f\circ g$ is a PL map. Is the converse true? i.e. ...
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77 views

Construction of the shift map

Is there a standard way to construct the shift map on an infinite product or coproduct of a direct or inverse system of spectra that induces the standard shift map of abelian groups in homology? Is it ...
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106 views

Homology of subsets of $\mathbb R^n$

Let $E \subset \mathbb R^n$. Must the homology groups $H_k (E)$ be trivial for $k \geq n$? How about just for $k > n$? If not, whats an example? Thanks.
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Equivariance of a map from tom Diecks book

in the book "Transformation Groups" of tom Dieck on page 123 ff., an equivariant version of the Hopf classification theorem is developed. I extract the relevant data to state my question: (U, B) is a ...
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63 views

Proof of: $P^2$: $U_i\cap V_{i-1}$ empty or disconnected

I have to learn for a very important test on monday but I don't get along with the following exercise from Bredon's book »Topology and Geometry«: Let the projective plane $P^2=U_1\cup … \cup U_n$ ...
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131 views

Local contractibility of CW complex

I was trying to understand the notion of CW complex from wikipedia. The very first non-example is: $$\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subset \mathbb R^2$$ This is not ...
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82 views

A theorem by Hopf on surfaces

I am reading a paper on Riemann surfaces and faced a problem about one of the refernces the author gave in the exlaination of one of the results. Here is a summary of what I am reading. Let $X_1$ ...
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227 views

Homology groups of a 2-sphere with q cross caps

Fraleigh(7ed) Section43 Exercise9 The below is the full solution. But I can't understand the red underlined parts. 1) Why the piece that contains the i-th crosscap have boundary $z_i-2a_i$? I ...