## Geometric topology – non-positive section curvature in a three-dimensional manifold

The answer to the following question may be well known in the field of geometric topology, so I ask for help here.

Does the total space of the circle beam on a closed hyperbolic surface admit a Riemannian metric with non-positive section curvature?

In particular, the circular beam that comes from the tangent beam of the hyperbolic surface is directly related to my thesis on the positive scalar curvature.

Since it does not have metrics with non-positive section curvature, it may not be of an expandable variety, which is a barrier to admission. metrics of positive scalar curvature.

## network topology – Calculation of the new router table C?

Consider the subnet as shown below, where the distance vector routing is used and the following vectors have just arrived on the router C: B (5, 0, 8, 12,6,2 ); from D (16, 12.6, 0.9, 10); and E (7, 6, 3, 9, 0.4). The delays measured at B, D and E are respectively 6.3 and 5 (10). What is the new C routing table?

I do not know how to solve this problem. help me please solve this problem.

## General topology – Restrictions of a local diffeomorphism

I wonder if a local dieleomorphism has the following property (prove or disprove):

Let $$M, N$$ to be differentiable varieties, and $$f: M to N$$ to be a local diffeomorphism. assume $$Z$$ is a closed subset in $$N$$ such as the restriction
$$f | _ {f ^ {- 1} (Z)}: f ^ {- 1} (Z) to Z$$
is a bijection. Then there is an open neighborhood $$U$$ of $$f ^ {- 1} (Z)$$ in $$M$$ such as the restriction $$f | _U: U to f (U)$$ is a diffeomorphism.

Thank you!

## General Topology – About Bernstein Sets

Remember that a subset $$X subseteq mathbb {R}$$ is a $$G _ { delta}$$-set if $$X$$ is a countable intersection of open sets in $$mathbb {R}$$. For example, closed subsets of $$mathbb {R}$$ are $$G _ { delta}$$-sets. A set $$Y subseteq mathbb {R}$$ is dense in itself if each point of $$Y$$ is a limit point of $$Y$$.

Also a set $$B subseteq mathbb {R}$$ is called a Bernstein Ensemble if $$P cap B not = emptyset$$ and $$P cap ( mathbb {R} setminus B) not = emptyset$$ for each uncountable closed set $$P subseteq mathbb {R}$$. Note that if $$B$$ is a set of Bernstein then $$mathbb {R} setminus B$$ is also a set of Bernstein.

Let $$B subseteq mathbb {R}$$ to be a Bernstein ensemble we have the following properties:

1. $$| B | = mathfrak {c}$$
2. $$B$$ There are no isolated points
3. $$B$$ is dense
4. $$B$$ is a bairoom
5. Yes $$G subseteq mathbb {R}$$ is a dense $$G _ { delta}$$ put, then $$B cap G$$ is dense in $$G$$ and $$( mathbb {R} setminus B) cap G$$ is dense in $$G$$.
6. Also if $$G subseteq mathbb {R}$$ is a dense in itself $$G _ { delta}$$ put, then $$G cap B not = emptyset$$ and $$G cap ( mathbb {R} setminus B) not = emptyset$$

My question is this: Someone knows more examples of subsets of the actual line satisfying properties 5 or 6.

Thank you

## At.algebraic topology – Homotopic type of linear isometric auto-isomorphisms of \$ R ^ infty \$

In the study "Orbispaces, Orthogonal Spaces and the Universal Compact Lie Group" by Stefan Schwede, he studies (spaces with an action of) the topological monoid. $$mathbf {L} ( mathbb R ^ infty, mathbb R ^ infty)$$ of the linear isometric self-nesting of $${ mathbb R} ^ infty$$ equipped with the subset topology of $$operatorname {maps} ( mathbb R ^ infty, mathbb R ^ infty)$$ with compact open topology. The underlying space of this monoid is contractable (Note A.12). What is known about the homotopic type of the subgroup of invertible elements of this monoid, ie the isometric linear isomorphisms of $${ mathbb R} ^ infty$$? In particular, is the underlying space always contractible?

## pr.probability – Low convergence in skorohod topology

Let $$D ((0, T); R ^ d)$$ either the cadlag function space with the usual Skorohod topology. $$X_t ( omega): = omega (t)$$ denotes the usual canonical process. Suppose that a family of probability measure $$mu ^ n$$ sure $$D ((0, T); R ^ d)$$ is tight with a low limit $$mu$$. So, is it true that for any bounded and continuous function $$f$$, we have
$$lim_ {n to infty} E ^ { mu ^ n} left ( int_0 ^ Tf (X_r) dr right) = E ^ { mu} left ( int_0 ^ Tf (X_r) dr right) ??$$
Or are there any references for that? Thank you very much.

## general topology – Assumption of Absorbent Sets in the Topological Vector Theorem

My question concerns this continuity of multiplication in a topological vector space. In concrete terms, it seems to me that the answer continues to be correct if we consider $$W = N times (-1,1)$$. However, in this case, I do not think the games are absorbing, because step 3 of the answer is a consequence of the fact that the games are balanced.

## Topology at.algebraic – For what types of \$ k \$ are \$ E_ {n, m} \$ – algebras automatically \$ E_ {n + 1} \$ algebras?

Recall that one $$E_ {n, m}$$ Algebra is a $$A_m$$ algebra in $$E_n$$ algebra. Here I index my $$A_m$$ algebras so that's one $$A_1$$ algebra is pointed, a $$A_2$$ Algebra has a unitary multiplication, $$A_3$$ is homotopic associative, etc. In particular, a $$E_n$$ Algebra is on the one hand a $$E_ {n, 1}$$ algebra and on the other hand a $$E_ {n-1, infty}$$ algebra.

Question: Suppose $$X$$ is a homotopy $$k$$-type and a $$E_ {n, m}$$Algebra when done
he has a canonical $$E_ {n + 1}$$ structure? In other words, how
connected is the opera map of the $$E_ {n + 1}$$-operated
$$E_ {n, m}$$-operad? And what would happen if $$m = 2$$?

Let me also briefly explain where this question comes from and why I am particularly interested in $$m = 2$$ Case.

A well-known theorem about monoidal categories says:

Theorem: a monoidal category $$mathcal {C}$$ is braided if there is a monoidal duplication of the forgetful canonical map of the center of Drinfeld $$Z ( mathcal {C}) rightarrow mathcal {C}$$.

I wondered what is the appropriate generalization of this statement in general $$E_n$$ algebra. This is for whom $$k$$-types make $$E_ {n + 1}$$ structures on $$E_n$$ the algebras correspond to the homotopic divisions of the forgetful map of the $$E_ {n + 1}$$ center $$Z (A) rightarrow A$$. If you think through the $$E_0$$ In this case, it is not difficult to see that a homotopic retraction of a $$E_1$$ Algebra is only one $$A_2$$ algebra. Apply this observation to $$E_0$$ algebras in $$E_n$$ algebras we should not expect to $$A$$ as above to be a $$E_ {n, 2}$$ algebra. So, the above theorem corresponds to the assertion that a homotopy $$1$$-type that is a $$E_ {1,2}$$ Algebra is automatically a $$E_2$$ algebra. Basically, it's because of the Eckman-Hilton argument, which says that the two multiplications agree and that the second multiplication is $$A_ infty$$ and not only $$A_2$$.

## Algebraic Topology – Homotopy of Chain Cards

Let $$A$$ and $$B$$ to be chain complexes. Let $$I$$ to be the complex of the chain with $$I_0 = mathbb {Z} (a, b)$$ (the $$mathbb {Z}$$ linearization of the whole $${a, b}$$) $$I_1 = mathbb {Z} (e)$$ with all the others $$I_n$$ trivial, and $$partial_1 (k cdot) = k cdot a + k cdot b$$.

Let $$H: I otimes A rightarrow B$$ to be a map of the chain I want to show that $$H ^ a: A rightarrow B, H ^ a (x) = H ( otimes x)$$ and $$H ^ b: A rightarrow B, H ^ b (x) = H (b otimes x)$$ are also chain cards.

Attempt:
Show $$H ^ a$$ is a chain map we have to show that $$partial ^ B_n circ H ^ a_n = H ^ a_ {n-1} circ partial ^ C_n$$. Since $$H$$ is a map of the chain we know $$partial ^ B_n circ H_n = H_ {n-1} circ partial ^ {I otimes C} _n$$. But I do not know how to continue

## General topology – The cellularity of the composition of cellular maps (with arbitrary CW decomposition)

Is the composition of cellular cards cellular?

In connection with this, I have another question. (I'm sorry to ask a very similar question.)

Let $${ sf CWcpx}$$ to be the category of CW complexes and leave $${ sf Top}$$ to be the category of topological spaces. Take a functor (natural) $$i: { sf CWcpx} to { sf Top}$$.

My question is "Does the composition $$g circ f$$ in the image of $$i$$ while $$f$$, $$g$$ are in the image of $$i$$? ".

More precisely, $$X$$,$$Y$$,$$Y & # 39;$$,$$Z$$ to be CW complexes such as $$Y$$ and $$Y & # 39;$$ are homeomorphic as topological spaces. And let $$f: X to Y$$, $$g: Y & # 39; to Z$$ to be cellular cards. My question (reformulated version) is: "Is there a pair $$(X, Z)$$ complexes and the cellular map $$h: X & # 39; to Z & # 39;$$ such as $$X cong X$$, $$Z cong Z$$ as topological spaces and $$h$$ is equal to $$g circ f$$? ".