## gn.general topology – On Čech-complete space

I’m reading an article of topology and i came across a Properties :

Properties :

Closed subspaces and arbitrary products of Čech-complete spaces are Čech-complete

Every Čech-complete space is a Baire space

Where the author just explain the concept. I am stuck on what actually does it mean and how to prove it. Is there an easy way to prove this propeties?

## general topology – A topological detail in the definition of Lie group

A Lie group $$G$$ is an $$r$$-times-differentiable manifold endowed with a group structure, i.e. with an associative binary operation

$$mu:quad Gtimes G longrightarrow G :qquadleft{x,, yright} longmapsto xcenterdot y$$

and an inversion operation

$$zeta:quad G longrightarrow G :qquad g longmapsto g^{-1}~~,$$

both of which are $$r$$-times-differentiable.

Does this definition restrict in any way the topology wherewith the manifold is equipped?

More specifically, if I take the inverses of all points residing in an open set $$U$$,
$$tilde{U}equivzeta(U)~~,$$
will the resulting set $$tilde{U}$$ be open in that topology?

## gn.general topology – The space of skew-symmetric orthogonal matrices

Let $$M_n subseteq SO(2n)$$ be the set of real $$2n times 2n$$ matrices $$J$$ satisfying $$J + J^{T} = 0$$ and $$J J^T = I$$. Equivalently, these are the linear transformations such that, for all $$x in mathbb{R}^{2n}$$, we have $$langle Jx, Jx rangle = langle x, x rangle$$ and $$langle Jx, x rangle = 0$$. They can also be viewed as the linear complex structures on $$mathbb{R}^{2n}$$ which preserve the inner product.

I’d like to understand $$M_n$$ better as a topological space, namely an $$(n^2-n)$$-manifold.

$$M_1$$ is just a discrete space consisting of two matrices: the anticlockwise and clockwise rotations by $$pi/2$$.

For $$n geq 2$$, we can see that $$M_n$$ is an $$M_{n-1}$$-bundle over $$S^{2n-2}$$. Specifically, given an arbitrary unit vector $$x$$, the image $$y := Jx$$ must lie in the intersection $$S^{2n-2}$$ of the orthogonal complement of $$x$$ with the unit sphere $$S^{2n-1}$$. Then the orthogonal complement of the space spanned by $$x$$ and $$y$$ is isomorphic to $$mathbb{R}^{2n-2}$$, and the restriction of $$J$$ to this space can be any element of $$M_{n-1}$$.

Since the even-dimensional spheres are all simply-connected, it follows (by induction) that $$M_n$$ has two connected components for all $$n in mathbb{N}$$, each of which is simply-connected. For instance, $$M_2$$ is the union of two disjoint 2-spheres: the left- and right-isoclinic rotations by $$pi/2$$. The two connected components of $$M_n$$ are two conjugacy classes in $$SO(2n)$$; they are interchanged by conjugating with an arbitrary reflection in $$O(2n)$$.

Is (each connected component of) $$M_n$$ homeomorphic to a known well-studied space? They’re each an:

$$S^2$$-bundle over an $$S^4$$-bundle over $$dots$$ an $$S^{2n-4}$$-bundle over $$S^{2n-2}$$

but that’s not really very much information; can we say anything more specific about their topology?

## at.algebraic topology – \$p\$-completeness of the function spectrum \$F(Sigma^{infty} BS, Sigma^{infty} BK)\$

Let $$S$$ be a finite $$p$$-group and $$K$$ a compact Lie group, in the paper A Segal conjecture for $$p$$-completed classifying spaces, it is said that the function spectrum $$F(Sigma^{infty} BS, Sigma^{infty} BK)$$ is $$p$$-complete, but I have not succeeded in proving it. I hope this remains true when, more generally, $$S$$ is a $$p$$-toral group. Any suggestion or idea?, please.

## differential topology – May this slice disk for the unknot be pushed into the boundary?

Write the 4-ball as $$mathbb{D}^4=mathbb{D}^2times mathbb{D}^2$$.
Then its boundary $$mathbb{S}^3simeq mathbb{S}^1times mathbb{D}^2cup mathbb{D}^2times mathbb{S}^1$$. We will use implicitely this homeomorphism.

Consider the unknot $$K=mathbb{S}^1times{0}$$ in the boundary $$partial mathbb{D}^4simeqmathbb{S}^3$$ of the four ball.

The disk $$D= mathbb{D}^2times {0}$$ is a smooth slice disk for $$K$$.

Is $$D$$ boundary parallel? I.e. is it obtained by pushing an unknotting disk $$D’subset mathbb{S}^3,$$ $$partial D’ = K$$ inside $$int(mathbb{D}^4)?$$

A possible approach: take as $$D’subset partial mathbb{D}^4$$ the PL disk obtained by gluing the annulus $$mathbb{S}^1times ((-1,1)times {0})in mathbb{S}^1times mathbb{D}^2subsetpartial mathbb{D}^4$$ to the disk $$mathbb{D}^2times ({-1}times {0})in mathbb{D}^2times mathbb{S}^1subsetpartial mathbb{D}^4$$.
Then gluing $$D’$$ to $$D$$ we get an embedded 2-sphere. If we manage to prove that this sphere is unknotted, i.e. it bounds a 3-ball then we can use the latter to push $$D’$$ to $$D$$.

Relevance of this problem
In studying Kirby calculus, one finds often the claim that when you attach a 2-handle, the cocore of a 2-handle is an unknotted 2-disk, i.e. boundary parallel. How to prove this?
The above problem is a possible way.

## gn.general topology – irreducible compact set vs prime compact set

Let X be a topological space.
A compact set K is called irreducible if for any two compact subsets K1,K2 of K with K is equal to the union of K1 and K2, then K is equal to K1 or K2.
A compact set K is called prime if for any two compact sets K1,K2 of X with K is included in the union of K1 and K2, then K is included in K1 or K2.

Are these two properties equivalent?
I guess that they are not the same. But I could not come up with an example.

## ag.algebraic geometry – Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in \$n\$ variables and up to degree \$d\$)

Set $$x=(x_{1}, dots, x_{n}).$$ Consider the set $$mathbb{R}(x)_{d}$$ of polynomials with coef. in $$mathbb{R}$$ in $$n$$ variables up to degree $$d.$$ This set can be seen as a finite-dimensional vector space with basis formed by all the monomials up to degree $$d$$ in $$n$$ variables and thus endowed accordingly with a(n Euclidean) topology coming naturally from this vector space structure. Now consider particularly the subset $$mathbb{R}0(x)_{d}$$ of polynomials in $$mathbb{R}(x)_{d}$$ that, when restricted to a real line, have only real zeros (see Section 2.1 here where RZ polynomials are defined). Finally, take the subset $$bmathbb{R}0(x)_{d}subsetneqmathbb{R}0(x)_{d}$$ of polynomials defining bounded real algebraic varieties. Does $$bmathbb{R}0(x)_{d}$$ has non-empty interior in $$mathbb{R}(x)_{d}$$ with respect to the Euclidean topology previously mentioned?

I really intuit that this question has to be easily answerable and that $$bmathbb{R}0(x)_{d}$$ has in fact non-empty interior in $$mathbb{R}(x)_{d}$$ but I cannot see how to prove it right now. Thanks for any insight!

## general topology – Discussing the definition of continuity of linear functionals on the space \$mathcal{D} (mathbb{R}^{n})\$

$$mathcal{D} (mathbb{R}^{n})$$ the vector space over the field $$mathbb{C}$$ such that its elements are “nice” functions $$varphi :mathbb{R}^n to mathbb{R}$$. We say that a function is “nice” if:

1. It’s smooth
2. It has compact support

This is what I’ve studied.

The functional $$F_{f} (varphi)$$ is continuous in the following sense: Consider a sequence of functions $${ varphi_{k}(x) }$$ in $$mathcal{D} (mathbb{R}^{n})$$ with the following two properties:

1. There exists a compact set $$K subset mathbb{R}^{n}$$ such that for all $$k$$, $$mbox{supp} varphi_{k} subset K$$.

2. $$lim_{k to infty} partial^{m}varphi_{k}(x) = 0$$ uniformly for all $$m = 0, 1, 2, cdots$$.

Such a sequence is said to go to $$0$$ in $$mathcal{D}$$ and is written $$varphi_{k} to 0, k to infty$$ in $$mathcal{D}$$.

We then say that the functional $$F_{f}(varphi)$$ is continuous if $$F_{f}(varphi_{k}) to 0$$ for $$varphi_{k} to 0, k to infty$$ in $$mathcal{D}$$.

Continuity is a topological property. The space $$mathcal{D}$$ is a linear vector space. It is made into a topological vector space by defining the neighbourhood of the “point” $$varphi (x) = 0$$ by a sequence of semi-norms.

I’ve read that the two conditions required above in the definition of $$varphi_{k} to 0$$ in $$mathcal{D}$$ follow from the conditions used to define the neighbourhood of $$varphi(x) = 0$$. Why is this true though?

## at.algebraic topology – unlinking when relaxing the homeomorphism condition

Say that we have two knots $$K_1$$ and $$K_2$$ in $$S^3$$ linked together in $$S^3$$ and forming the Hopf link. Usually, we can prove that we cannot unlink them by using a link invariant that shows that the “two-component unlink” that consists of two separate circles in $$S^3$$ have a different value (with respect to the invariant) in comparison to its value on the Hopf link. This effectively shows that there is no homomorphism from $$S^3$$ to itself that separates the two links. I want to relax the condition of homomorphism a little bit and ask: is there a continuous function that separates the images of the two links? in other words, is there a continuous function $$f:S^3to S^3$$ such that $$f(K_1)$$ is contained in an closed disk $$D_1$$ and $$f(K_2)$$ is contained in another closed disk $$D_2$$ and $$D_1$$ and $$D_2$$ are disjoint? it seems that the answer is no but I am not sure how to show something like this. Any pointer is appreciated.

## at.algebraic topology – Homotopy groups of certain geometric fixed point spectrum

Let $$G$$ be a finite group and $$E$$ be a genuine $$H$$-spectrum for $$Hleq G.$$ Then for any subgroup $$K$$ of $$G$$, consider the $$K$$-spectrum $$X=Res^G_K Ind^G_H(E).$$

Is there any reference for computing the homotopy groups of the geometric fixed point $$Phi^K(X): = (widetilde{Emathcal{P}}wedge X)^K?$$ Here $$widetilde{Emathcal{P}}$$ is a $$K$$-space given the following fixed point data:

$$widetilde{Emathcal{P}}^L= begin{cases} S^0, & text{ if }L=K\ ast, & text { if } L text{ is any proper subgroup of } K.end{cases}$$
Maybe the double coset formula could be useful.

Thank you so much in advance. Any help will be appreciated.