general topology – What are the caps in the proof of Poincare Conjecture and does the insertion of caps into initial manifold preserve homeomorphism?

Quote from Wikipedia article “Poincare Conjecture”:

“He wanted to cut the manifold at the singularities and paste in
caps (Question), and then run the Ricci flow again…
In essence, Perelman showed that all the strands that form can be
cut and capped (Question)…”

To my notice, the above “Question” is: the caps do not belong to the
original manifold; thus, there is no direct (one to one) correspondence between
the original manifold and the final sphere. In conclusion, this formulation
violates the Poincare Conjecture:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Should somebody explain this point in Wikipedia? The public deserves
to read a good article on Wikipedia.

at.algebraic topology – Are category of perfect groups and acyclic space equivalent?

Here is my approach to relate two subcategories of $textbf{hTop}$ and $textbf{Gp}$ respectively:

Let $mathcal{C}$ denote the full subcategory of $textbf{hToP}$ consisting of acyclic( i.e. reduced cyclic homology in all degree is zero ) CW complex and $mathcal{D}$ denote the category of perfect groups (i.e. commutator is itself). There is functor $mathcal{F} = pi_{1} : mathcal{C} rightarrow mathcal{D}$ takes space $X$ to its fundamental group $pi_{1}(X)$. We have also another functor $mathcal{G} : mathcal{D} rightarrow
mathcal{C}$
given as if $P$ is any perfect group then define $mathcal{G}$(P) to be the homotopy fiber of the plus construction map $ BP rightarrow BP^{+}$. The above two functors seems too natural to me. So mine natural questions are the following:

  1. Are $mathcal{C}$ and $mathcal{D}$ equivalent categories?
  2. Is $mathcal{F}$ related to $mathcal{G}$ (I mean whether they are in adjunction, equivalence pair etc.)?

I computed the composition $mathcal{FoG}$: Let $P$ be a perfect group and $F(f)$ denote the homotopy fiber of $ BP rightarrow BP^{+}$. Then $mathcal{FoG}(P) = pi_{1}(F(f))$ and the counit map $mathcal{FoG}(P) = pi_{1}(F(f)) rightarrow P $ is the universal central extension map which in particular says that if we again repeat the same process turns out to be identity. So this is in a way saying that $mathcal{FoG}$ may not be identity its square is identity. I don’t know what it means. Any help would be appreciated.

gt.geometric topology – General Construction of Fuchsian Groups(finitely generated as well as infinitely generated) whose limit set is a cantor set

I have already studied the following books and PDF ;
(1) Hyperbolic Geometry, James W. Anderson, ( first five chapters )
(2) Fuchsian Group , Svetlana Katok. (first four chapters )
(3) Hyperbolic Geometry , MA 448 Caroline Series.
(4) Hyperbolic Geometry Charles Walkden MATH 32052.
(5) Hyperbolic Geometry, Jonathan M. Fraser, MT5830.
(6) The Geometry of Discrete Groups, Alan F. Beardon, (Chapter- 5, 8,9 )
(7) Geodesic and Horocyclic Trajectories, Francoise Dal’Bo (first two chapters)
(8) Indra’s Pearls, D. Mumford, C. Series. (first six chapters )
(9) An Introduction to Geometric Topology, Bruno-Martelli. (first six chapters ).
Actually I am trying to construct the general form of Fuchsian Groups whose limit set is a cantor set. Reading the above books I am thinking that the general form of Fuchsian Groups whose limit set a cantor set if it corresponds to a surface with at least one funnel (boundary component has no cusps). Also, reading the book of Bruno- Martelli, I am further thinking that what I actually need is basically a hyperbolic surface with infinite type . That’s why nowadays I am reading the book of L. Ahlfors, L. Sario, Riemann Surfaces.
Further I am thinking that what I need is infinite dimensional teichmullar space (space of all fuchsian group of second kind ), for this I have seen the paper of Alastair Fletcher and Vladimir Markovic, Infinite dimensional Teichm¨uller spaces, and also seen the book Volume 1 Teichmiiller Theory
John Hamal Hubbard.
But unfortunately right now I could not progress my research problem . Would you please share your valuable thought on my research problem ? Am I going to right direction or not ? Please tell me that.
Also, would you please suggest me some books or pdf (apart from those which I have studied already) that will very great full to me.

general topology – If $f$ is bijection in a dense subset then $f$ is bijection in all space

Let $X=(X,mathcal{T}_X)$ and $Y=(Y,mathcal{T}_Y)$ be topological Hausdorff spaces and $f: X longrightarrow Y$ be a continuous function. If $f:D subset X longrightarrow Y$, with $D$ dense in $X$, is a bijection (one-to-one and onto) then $f:X longrightarrow Y$ is a bijection too.

This is true in general?

at.algebraic topology – Cellular chain complex of $G$-CW-complexes & their differentials

I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of
$G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf

Recall, the cooking recipe for cellular homology
in case classical CW-complexes without any $G$-structure.

$X$ is $k$-dimensional CW-complex if
we have a decomposition $X_0 subset X_1 subset … subset X_{n-1} subset X_n=X$
where every $X_{i+j}$ arises as pushforward of $X_j$
after attatching $I_j$ many $j$ cells $cong D^j$ along $S^{j-1} subset D^j$ to $X_j$.
The we build a complex

$$C_k(X) to C_{k-1}(X) xrightarrow{text{d}} … to C_1(X) xrightarrow{text{d}} C_0(X) to 0$$

where $C_j := bigoplus_{i in I_j} mathbb{Z}$ is free $mathbb{Z}$-module of rank $I_j$ and $d$-degree maps. Then compute homology
of this complex.

My question is if we now would like to compute cellular homology of a $G$-CW-complex
how the $C_j(X)$ would look like. Would they also be free $mathbb{Z}$-modules or
will they have additional structure? Maybe as $mathbb{Z}G$-modules?
The reason why I worry about this is following example 2.31:

enter image description here

Consider EXAMPLE 2.31. We deal with finite cyclic group $G:= mathbb{Z}/n$ that act on circle $S^1$ considered
as $G$-CW-CW-complex with $n$ vertices ($0$-cells) and $n$ $1$-cells.
The obviuos action by $G$ is that the generator $t := bar{1}$ of
$G$ shift the $1$-cells resp
$0$-cells conterclockwise:

enter image description here

Then in the example following complex is considered:

$$mathbb{Z}G to mathbb{Z}G to mathbb{Z} to 0$$

Is this the analogon of cellular complex $C_k(X) xrightarrow{text{d}} … to C_0(X) to 0$ from above? Then the message is $C_1(X)=C_0(X)= mathbb{Z}G$?

Set theoretically that’s fine since the number of $1$– and $0$-cells coinsides
with $vert G vert$ but as what are $C_j(X)$ and the cellular complex are considered?

As $mathbb{Z}$-modules and $mathbb{Z}G$-structure play no role for homology or as a proper $mathbb{Z}G$-complex? And what are the differentials $d$ here? Maps of $mathbb{Z}$ or $mathbb{Z}G$-modules?

More generally, let $X$ is $k$-dimensional $G$-CW-complex if
we have a decomposition $X_0 subset X_1 subset … subset X_{n-1} subset X_n=X$
of $G$-cells and assume that this $G$-CW has $I_j$ many $j$ cells.

What is $C_j(X)$ explicitly and again what are the $d$-maps of the corresoponding complex

$$C_k(X) xrightarrow{text{d}} C_{k-1}(X) xrightarrow{text{d}} … to C_1(X) xrightarrow{text{d}} C_0(X) to 0$$

Are they maps of $mathbb{Z}$ or $mathbb{Z}G$-modules?

general topology – Are these characteristic functions Lebesgue measurable?

I’m trying to prepare myself for the final exam in real-analysis and I received this problem from my professor, could you help me, please?
Let $Esubseteq(0,1)$ s.t. $E notin mathbb{L}(mathbb{R}), $A=(-1,1)E$ and let $chi_A$, $chi_E$:$mathbb{R}to(0,1)$ be the corresponding characteristic functions.<br /> Determine if the functions $chi_A$, $chi_E$, $chi_A$+$chi_E$, $chi_A$-$chi_E$ are Lebesgue measurable.

at.algebraic topology – Oddness of intersection form of surface bundle

Let $Sigma_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle: $Sigma_g to M^4 to Sigma_h$. When $g=1$, $M^4$ is called a torus bundle.

My question: is there a torus bundle whose intersection form contains an odd diagonal element (if we choose a basis and view the intersection form as a matrix)?

If $M^4=Sigma_1times Sigma_h$, then $M^4$ is spin and its intersection form has only even diagonal elements.

More generally:
For a given fiber $Sigma_g$, is there a $Sigma_g$-bundle whose intersection form is odd?

general topology – Continuity on intersection of open sets

f :X ->Y is a mapping of a metric space X into a metric space Y, and A and B are open subsets of X. Prove that, if f is continuous on A and on B, it is continuous on A intersection B. Is this result true if A and B are closed subsets? Is the result true for the union of (i) an infinite number of open sets(ii) an infinite number of closed sets

gt.geometric topology – Is there an orientable prime manifold covered by a non-prime manifold?

A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere.

Is there an example of a finite covering $pi : N to M$ of closed orientable manifolds where $M$ is prime and $N$ is not?

There are no examples in dimensions two or three. If one is willing to forgo the orientability requirement, then there are examples in dimension three. In this paper, Row constructs infinitely many topologically distinct, irreducible (and hence prime), closed 3-manifolds with the property that none of their orientable covering spaces are prime.

There are examples where $N$ is prime and $M$ is not, such as the double covering $pi : S^1times S^2 to mathbb{RP}^3#mathbb{RP}^3$.

general topology – Proof of a Topological Embedding

I encountered a theorem from the Book Elementary Topology: Problem Book:

Theorem. Let $p:Xto Z$ be a closed quotient map where $X$ is compact Hausdorff. Then for every topological space $Y$, the function $Phi:C(Z,Y)to C(X,Y)$ defined by $fmapsto fcirc p$ is a topological embedding.

Here $C(Z,Y)$ and $C(X,Y)$ are assumed to be given with the compact-open topology.

It is easy to prove thet $Phi$ is continuous and injective:

Proof of Continuity. Let $S(C,U)$ be a subbase element of $C(X,Y)$ where $C$ is compact in $X$ and $U$ is open in $Y$. Since the quotient map $p$ is continuous, $p(C)$ is compact in $Z$. Then we claim that
$$Phi^{-1}(S(C,U))=S(p(C),U)subseteq C(Z,Y).$$
On the one hand, for each $finPhi^{-1}(S(C,U))$, we have
$$f(p(C))=(fcirc p)(C)=Phi(f)(C)subseteq U.$$
On the other hand, for each $fin S(p(C),U)$, we have
$$Phi(f)(C)=(fcirc p)(C)=f(p(C))subseteq U.$$
The equality thus follows. Note that $S(p(C),U)$ is a subbase element of $C(Z,Y)$, hence is open. Thus $Phi$ is continuous.

Proof of Injectivity. Moreover, for every $f,gin C(Z,Y)$, if $fneq g$, then $f(z)neq g(z)$ for some $zin Z$. Since $p$ is surjective, we have $z=p(x)$ for some $xin X$. Note that
begin{equation*}
Phi(f)(x)=(fcirc p)(x)=f(p(x))=f(z)
end{equation*}

and similarly, $Phi(g)(x)=g(z)$, so $Phi(f)neqPhi(g)$, implying that $Phi$ is injective.

Finally, it suffices to prove that $Phi$ is an open map. I followed the hint of the book:

Hint. Let $Ksubseteq Z$ be a compact set, $U$ open in $Y$. The image of the open subbase set $S(K,U)subseteq C(Z,Y)$ is the set of all maps $g:Xto Y$ constant on all $p^{-1}(z)$ where $zin K$ and such that $g(p^{-1}(K))subseteq U$. It remains to show that the set $S(p^{-1}(K),U)$ is open in $C(X,Y)$. Since the space $Z$ is Hausdorff, it follows that the set $K$ is closed. Therefore,
the preimage $p^{-1}(K)$ is closed, and hence also compact. Consequently, $S(p^{-1}(K),U)$ is a subbase set in $C(X,Y)$.

The hint is too redundant, so I add more details about that:

First, since $X$ is compact Hausdorff, it is clear that $X$ is normal Hausdorff (both $T_1$ and normal).

The quotient map $p$ is closed, so $Z$ is also normal Hausdorff.

Note that $K$ is compact in $Z$, so $K$ is closed. As a result, the set $p^{-1}(C)subseteq X$ is closed and hence compact.

The set $S(p^{-1}(K),U)$ is open in $C(X,Y)$ because it is a subbase set.

But here is a problem: We can only obtain that
$$Phi(S(K,U))subseteq S(p^{-1}(K),U)$$

The hint also mentioned that

The image of the open subbase set $S(K,U)subseteq C(Z,Y)$ is the set of all maps $g:Xto Y$ constant on all $p^{-1}(z)$ where $zin K$ and such that $g(p^{-1}(K))subseteq U$.

It seems hopeless to deduce $Phi(S(K,U))$ is open in $C(X,Y)$ by these arguments.

I am not sure whether I have missed something, but I still do not think such argument would go. Hope anyone had good ideas on this.