differential topology – Transversality theorem for maps between fiber bundles

I am looking for a possible generalization of the standard Trasversality Theorem which roughly says that transverse maps are generic. For example, see the version below:
From page 74, Theorem 2.1 in Hirsch, Differential Topology, GTM 33

Let us modify the step-up to a more general situation. Now assume furthermore that $M,N$ are both the total spaces of fiber bundles over the same base manifold $B$. It is possible that $M,N$ have different fibers over $B$. Everything else is the same as before. Let $C^infty(M,N,B)$ denote the space of smooth $f:Mto N$ making the following diagram commute (under strong or weak topologies, if it matters in this case):
$require{AMScd}$
begin{CD}
M @>f>> N\
@V V V @VV V\
B @= B
end{CD}

where the vertical maps are the bundle projections. Let $T(M,N,B;A)$ denote the subspace of those $f$ that are transverse to a submanifold $Asubseteq N$.

My question is: Are maps transverse to $A$ generic in any sense among those maps making the diagram commute? For example, is the subset $T(M,N,B;A)$ residual in $C^infty(M,N,B)$? Could anyone guide me to a reference please?

While my intuition might lead (or mislead?) me to believe that transverse maps are always generic, I am not sure if I need to make any additional assumptions on $A$ or on the fiber bundles $Mto B$ and $Nto B$. Notice that if $B$ is a single point, then the new step-up is the same as the old one. So I am indeed asking for a generalization of the usual transversality theorem here.

In a sense, the questions above ask for a “transversality theorem parametrized by $B$“. I am aware that there is a parametric transversality theorem which seems similar in spirit to what I want. But I don’t think it answers my questions above.

gn.general topology – Statements related to Thurston’s work on the surface

If we have α and β be simple closed curves on a surface Σg. The intersection number i(α,β) is defined to be the minimal cardinality of α1 ∩ β1 as α1 and β1 ranges over all simple closed curves isotopic to α and β, respectively. We say α and β intersect minimally if i(α, β) = |α ∩ β|.

How to see thatα and β intersect minimally if there are no pairs of p, q ∈ α ∩ β such that the arc joining p to q along α followed by the arc from q back to p along β bounds a disk in Σg? (maybe a sketch of the idea of proof?)

I think the converse is also true : “thatα and β intersect minimally only if there are no pairs of p, q ∈ α ∩ β such that the arc joining p to q along α followed by the arc from q back to p along β bounds a disk in Σg.”

general topology – Mapping a semi-infinite strip to the first quadrant

I want to find the image of the semi-infinite strip
$$
0 le Re(z) le fracpi2, Im(y) ge 0
$$

under the map
$$
w = frac{i}{sin z}
$$

I first rewrite the mapping function as
$$
w = frac{1}{cos x sinh y – i sin x cosh y}
$$

So the right boundary gets mapped to the imaginary axis from 0 to 1; the bottom boundary gets mapped to the imaginary axis from 1 to $infty$, and the left boundary gets mapped to the positive real axis. From here, it becomes clear that this maps the infinite strip to the first quadrant. But I don’t how to manipulate the points within the strip. e.g. the vertical lines. I tried to plug in $x = c$ into the mapping function, but don’t know how to further simplify it. Can someone give me a hint? Thanks!

gt.geometric topology – A certain property for Heegaard splittings

I’ve become interested in 3-manifolds with the following property (called ‘Property A’): let $c_{i}$ be a set of $g$ curves on a genus $g$ surface $Sigma$ and let $b_{i}$ be the $g$ meridional curves of $Sigma$. That is, if $Sigma$ is the boundary of a genus $g$ handlebody, then the $b_{i}$ bound compressing disks. Then ${c_{i}}_{i=1,dots,g}$ satisfy ‘Property A’ if, for any $b_{j}$, the geometric intersection numbers $iota(c_{i}, b_{j})$ have the same sign for all $i=1,dots,g$. Informally, all of the attaching curves for the 2-handles run over the 1-handles in the same direction.

So my question is, which 3-manifolds admit a Heegaard splitting with the attaching curves of the 2-handles satisfy Property A?

For example, any direct sum of lens spaces will satisfy this. As does the following Heegaard diagram for the Poincaré homology sphere (taken from Manifold atlas http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_homology_sphere):

A possible nonexample could be the 3-torus, for which the only Heegaard diagram I know does not satisfy this property.

Lastly, and of course this depends heavily on my first question, is it possible that all rational homology spheres satisfy this?

I would be grateful for any sort of insight.

at.algebraic topology – Maps between unitary little disks operads and non-unitary little disks operads

Derived mapping spaces between little $d$-disks operads $E_d$ play an important role in embedding calculus. For example, Dwyer-Hess expresses the homotopy of framed long knots as loop spaces such mapping spaces, a result which was generalized by Boavida de Brito-Weiss.

When stating these results, the $E_d$-operad is usually unitary: its space of 0-ary operations is contractible (in fact a point). There is also a non-unitary version $E_d^{nu}$, where replace the $0$-ary operations by an empty set. There is a natural forgetful map from the derived mapping space of unitary operads to that of non-unitary operads

$$Map^h_{mathsf{Op}}(E_m,E_n) to Map^h_{mathsf{Op}}(E_m^{nu},E_n^{nu}).$$

Is this a weak equivalence?

This is known if we assume $m=n$ and we replace the operads by their rationalizations, by Section 7 of Fresse-Willwacher.

general topology – Which characterization of continuity in a point is used here?

I am currently studying a proof, where we have a function $f: Ato Y$ where $Asubseteq X$ is a topological space, and $Y$ is a metric space.

We want to show that $f$ is continuous in a point $x$, and the proof goes along to show that for a specific open neighborhood of $f(x)$ the preimage contains an open neighborhood of $x$.

For $varepsilon >0$ we show that $f^{-1}(B_varepsilon(f(x)))$ contains an open neighborhood of $x$.
Which characterization of continuity of a point is then used here?

Do we not have to show that for EVERY open neighborhood $V$ of $f(x)$, the preimage contains an open neighborhood $U$ of $x$.

A function $f:Xto Y$ of topological spaces is continuous in $x$ if the preimage of neighborhoods of $f(x)$ or neighborhoods of $x$.

But in the given proof it looks like that for a general type of open neighborhoods this is shown.
So to general in my opinion.

Or is this just an equivalent characterization that I am missing here?

Thanks in advance.

self learning – In which order should I read Pete Clark’s general topology or what is the prerequisite for Pete Clark’s general topology?

Recently,I start to read Prof. Pete Clark’s lecture notes on general topology,and first I read the appendix,however I find it very hard for me to learn. There are some objects with no definition in the appendix.In fact,I am a self learner,I have only learned calculus and linear algebra,and know some basic real analysis objects,such as countable sets,uncountable sets,mappings,etc.Many people tell me that if you have knowledge of calculus,you can start to learn general topology.However,for now,I think that Pete Clark’s general topology is hard for me.

at.algebraic topology – Goodwillie derivatives of $X mapsto Sigma^infty X^{wedge n}/Sigma_n$

I’m following Arone’s lectures on Goodwillie calculus from Munster 2015. There he left an exercise:

Find $partial_kF$ for $F: text{Top}_* to text{Sp}$ given by $F(X) = Sigma^infty X^{wedge n}/Sigma_n$ where $X^{wedge n} = X wedge cdots wedge X$ and $Sigma_n$ is the symmetric group, which acts on $X^{wedge n}$ by permuting factors.

In the lectures he solves the case $n = 2$, that gives $partial_1 F simeq Sigma^inftySigmamathbb{RP}^infty$, $partial_2F simeq Sigma^infty$ and $partial_nF simeq *$.

I do not know how to work out the general case. I know that $partial_kF simeq text{hocolim }Omega^{km}text{cr}_kF(S^m, dots, S^m)$.

For $k = 1$ then $partial_1F simeq text{hocolim }Omega^mF(S^m)$. My idea is to write $F(S^m) = S^m wedge G(S^m)$ for some $G$ in order to cancel $Omega^m$. This can be done by choosing a $phi: (S^m)^{wedge n} to (S^m)^{wedge (n-1)}$ such that $(x_1, dots, x_n) mapsto (x_1 + cdots + x_m, phi(x_1, dots, x_m))$ is a homemorphism. In this case $(S^m)^{wedge n}/Sigma_n cong S^m wedge (S^m)^{wedge (n-1)}/Sigma_n$ where the $Sigma_n$-action on $(S^m)^{wedge (n-1)}$ depends on $phi$. Unfortunately, I was not able to find a $phi$ such that this action looks nice or familiar as in the case $n = 2$.

Alternatively, I tought $(S^m)text{^}^n/Sigma_n cong text{subset}_{leq n}(mathbb{R^m})^+ cong S^m wedge text{subset}_{leq n, 0}(mathbb{R^m})^+$
where $(cdots)^+$ means compactification, $text{subset}_{leq n}(cdots)$ is the space of subsets of cardinality $leq n$ and $text{subset}_{leq n, 0}(mathbb{R^m})$ is the space of those with barycenter 0. Then $partial_1F simeq Sigma^inftytext{subset}_{leq n, 0}(mathbb{R^infty})$. This agrees with the previous result for $n = 2$ since any two unordered points can be identified by the line joining them ($mathbb{RP}^infty$) and the distance from 0 ($Sigma$, after compactification). But

  1. It does not seem to be the “right solution”.
  2. What about $partial_nF$ for $n > 1$? I do not know how to proceed.

Thanks in advance!

(original post)

compactness – Show that if $X$ is uncountable and $(X,F)$ is the co-countable topology on $X$, is not compact.

Show that if $X$ is uncountable, and $F$ is the collection of sets $E$ such that $Xsetminus E$ is empty or at most countable. Then $(X,F)$ the co-countable topology on $X$, is not compact.

I know I’ve got to make a collection of sets in $F$ which don’t have a finite sub cover. I want to say I can get a collection $bigcup_{n=1}^{infty}V_n=X$ such that each $V_{1}^csupset V_{2}^csupset …$ are countable. Since if I then take only a finite collection of these, I’d have $(bigcup_{i=1}^n V_i)^c$ isn’t empty, as one of them has the smallest complement, which is non empty. So they don’t cover $X$. But I’m not sure how to make such a set.

Maybe I can just take any countable subset of $X$, call it $U$ and define $V_1$ as $Xsetminus U$, then define $V_2$ as $Xsetminus (Usetminus {u})$ for some $uin U$, and continue this by saying $V_i=Xsetminus(V_{i-1}setminus {u})$ for some $uin U$. Which I believe will cover $X$ eventually because any $xin X$ is either in $V_1$ at the start or is in $V_i=Xsetminus (V_{i-1}setminus {x})$ but I can’t cover $X$ with only finitely many since I’ll always miss some subset of $U$.

I’m not sure if I can actually construct such a set though.

gt.geometric topology – Harmonic maps versus Teichmuller maps between Riemann surfaces

Let $(X,phi)$ be an element of Teichmuller space $cal T_g$ and $q$ a (holomorphic) quadratic differential on $X$. Teichmuller geodesic flow gives a family of marked Riemann surfaces $(Y_t,psi_t) = exp(tq)$ for $tin mathbb R$. A quadratic differential $tq$ also determines a unique marked Riemann surface $(Y’_t,psi’_t)$ and harmonic map $u_t:Xto Y_t’$ described below. Notice that when $t=0$, $X=Y_0=Y_0’$.

My question is: how $Y_t$ and $Y_t’$ are related. For example, is there some bound on the distance between them?

To describe how to get a Riemann surface from a quadratic differential, I am summarizing the discussion from Section 4.2 in Jurgen Jost’s book Compact Riemann Surfaces:

Between any two points in Teichmuller space $(X,phi),(Z,psi)$ there is a unique harmonic map $u:Xto Z$ in the homotopy class of $psicirc phi^{-1}$. Let $rho^2dudbar u$ be the hyperbolic metric on $Z$. Then $(rhocirc u)^2 frac{partial u}{partial z}frac{partial bar u}{partial z}dz^2$ is a holomorphic quadratic differential on $X$ and this map $mathcal{ T}_g to QD(X)$ between Teichmuller space and quadratic differentials is a bijection. Thus, from a quadratic differential $q$ on $X$, we get an element of Teichmuller space.