Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
3
votes
0answers
36 views
Reference for couple facts in algebraic topology (tautological line bundle, principle G bundle)
I have seen the following couple of (basic and possible obvious) facts written when I look up sources on the internet, but I'm not so sure why they are true or where to find them.
1) We mentioned ...
3
votes
0answers
35 views
some help on the group of unknotted
Show that the group of the unknotted $K=\{(z_z,z_2)\in \mathbb{S^3} : |z_1|=1 \}$ is infinite cyclic. where $\mathbb{S^3}$ is to be considering as the unit vectors in $\mathbb{C^2}\cong \mathbb{R^4}$.
...
3
votes
0answers
41 views
Basic computation of a double graded spectral sequence: $^I E^0_{pq}$
Let $C=\oplus_{p\geq0, q\geq0}C_{pq}$ be a double graded group with two differentials: $d^I_{pq}:C_{pq}\rightarrow C_{p-1,q}$ and $d^{II}_{pq}:C_{pq}\rightarrow C_{p,q-1}$, with the usual assumption ...
3
votes
0answers
41 views
A short question on shriek maps
This should be easy but I don't quite see it. Let $M^m, N^n, X^d$ be compact, connected and oriented smooth manifolds. Let also $f:M\rightarrow X$ and $g:N\rightarrow X$ be transverse smooth maps. ...
3
votes
0answers
85 views
Simple exercise in cohomology
I know this is a simple exercise but I am stuck unfortunately.
Question:
Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
3
votes
0answers
81 views
Relation between complex and real sphere
I want to understand relation between complex and real spheres.
How to show?
$S^1(\mathbb{C}) \approx \mathbb{R} \times S^1$
$S^3(\mathbb{C}) \approx \mathbb{R} \times S^3$
$\approx$ means homotopy ...
3
votes
0answers
75 views
Topology Qualifying Exam Problem 42
I was going through some old qualifying exam problems and I have been struggling with this one. Any help would be great, thanks.
Consider the 2-dimensional torus $\mathbb{T}^2$ and the topological ...
3
votes
0answers
57 views
cellular chain complex of sphere
The cellular chain complex $C_{\ast}(X)$ of an $n$-sphere $X=S^{n}$ (with any CW-complex structure), gives rise to an exact sequence
$$ 0 \rightarrow \mathbb{Z} \rightarrow C_{n}(X) \rightarrow ...
3
votes
0answers
128 views
Mackey functor structure on equivariant homotopy groups
I have read that the equivariant stable homotopy groups $\pi_n^{-}(X)=\pi_n(X^{-}) $ of a $G$-space or $G$-spectrum $X$ have a Mackey functor structure.
Can somebody please explain how the covariant ...
3
votes
0answers
79 views
Groups not arising from certain centralizers
There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
3
votes
0answers
71 views
Classification of fundamental groups of non-orientable surfaces
I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$.
I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
3
votes
0answers
46 views
If $S_1$ is orientable and $S_2$ it isn't,
Let $S_1$ and $S_2$ be two closed surfaces. Demonstrate that the following conditions are necessary for there to be a $k$-sheeted covering $p: S_1 \rightarrow S_2 $.
a) $\chi(S_1)=k \chi (S_2)$.
b) ...
3
votes
0answers
90 views
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) ...
3
votes
0answers
35 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 ...
3
votes
0answers
87 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 ...
3
votes
0answers
38 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 ...
3
votes
0answers
31 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 ...
3
votes
0answers
57 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 ...
3
votes
0answers
72 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 ...
3
votes
0answers
78 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 ...
3
votes
0answers
69 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 ...
3
votes
0answers
150 views
Algebraic Morse theory
In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
3
votes
0answers
49 views
How to use the Prontrjagin-Thom construction to obtain the Gysin map?
I need help to understand the diagram in Miller's script Vector Fields on Spheres, etc. Chapter 23, p.82 on the bottom of the page.
Before, Miller introduces the Prontrjagin-Thom construction: It is ...
3
votes
0answers
61 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 ...
3
votes
0answers
189 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 ...
3
votes
0answers
32 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 ...
3
votes
0answers
90 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
0answers
85 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 ...
3
votes
0answers
65 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)$$
where $\Omega^n(X,x_0)$ is the $n$-loop group and $M(S^n,*; X,x_0)$ is the set of pointed continuous maps from $(S^n,*)$ to $(X,x_0)$
...
3
votes
0answers
48 views
homotopy type of the closure of a subset
Let $X$ be a topological space and $N$ a subset of $X$. Is it true that the closure of
$N$ in $X$ is homotopy equivalent to $N$. I think it is not. take for example $N=\mathbb Q\subset \mathbb ...
3
votes
0answers
110 views
cohomology isomorphism
Let $X$ be a finite dimensional CW complex and $A$ be a closed subset in $X$ and $N$ a regular neighborhood of $A$ that deformation retracts onto it. why do we have for each $i$,
$$H^{i}(X-A;\mathbb ...
3
votes
0answers
148 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 $r \circ i = id$?
3
votes
0answers
208 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 ...
3
votes
0answers
64 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 $p$ ...
3
votes
0answers
87 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 ...
3
votes
0answers
210 views
Torsion in homology groups of a topological space
It seems as though "nice" spaces don't have torsion in their homology groups. What is the underlying characteristic of these nice spaces; that they can be embedded in $\mathbb{R}^3$? So what are some ...
3
votes
0answers
84 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 ...
3
votes
0answers
96 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 ...
3
votes
0answers
147 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 ...
3
votes
0answers
204 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 ...
3
votes
0answers
94 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 ...
3
votes
0answers
119 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 ...
3
votes
0answers
159 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 ...
3
votes
0answers
104 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 ...
3
votes
0answers
198 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 ...
3
votes
0answers
255 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.
3
votes
0answers
55 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 ...
3
votes
0answers
298 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 ...
3
votes
0answers
65 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.
...
3
votes
0answers
72 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 ...
