Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

**3**

votes

**0**answers

162 views

### Straightening for $\infty$-operads

There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to ...

**1**

vote

**0**answers

79 views

### Loop Motion Planning Algorithms

Happy New Year.
In a similar spirit of question Motion planning algorithm, we consider a path connected topological space $X$, and equip its free loop space $X^{S^1}$ with the open compact topology. ...

**3**

votes

**1**answer

374 views

### homotopy groups of spheres. [closed]

I not sure that my question has a research level so feel free to remove it, I'll not be offended.
Let $S^{n}$ be a sphere of dimension $n>1$ and $p<q$ two prime numbers. Is there always a ...

**3**

votes

**2**answers

238 views

### Isomorphism between the Burnside ring $A(G)$ and the zeroth $G$-equivariant stable homotopy $\pi^{G}_0(S^0)$

Let $G$ be a compact Lie group. I know that the Burnside ring $A(G)$ is isomorphic to the zeroth $G$-equivariant stable homotopy $\pi^{G}_0(S^0)$. What is the isomorphism between $A(G)$ and ...

**9**

votes

**1**answer

272 views

### Example of a saturated class of morphisms which is not _obviously_ saturated?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists ...

**3**

votes

**1**answer

472 views

### Opposite Symmetric Monoidal Structure on an Infinity Category

Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of ...

**0**

votes

**1**answer

135 views

### relations between intersection form and Chern classes

Is there exist a 4-manifold which intersection form has the following property
$$
(a,a) \neq 0\ \text{if}\ a\neq 0,
$$
and the second (or the first) Chern class (for some almost complex stucture) ...

**3**

votes

**1**answer

137 views

### Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)

What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation?
By co-rank, I mean the ...

**26**

votes

**2**answers

1k views

### Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches:
Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...

**3**

votes

**1**answer

292 views

### Knots in 3-manifolds

Consider a closed $3$-manifold $M$ and a knot $K$ in $M$.
Is it necessarily true that $\pi_2 (M \setminus K) = 0$?
If not, are there any conditions on $M$ and/or $K$ to ensure the above 2nd homotopy ...

**4**

votes

**1**answer

286 views

### Punctured 3-manifold

Suppose we have a closed 3-manifold $M$, not necessarily simply connected.
What can I say about the homotopy groups of $M \setminus \text{pt}$? ($M$ punctured by one point)
In particular, what ...

**4**

votes

**1**answer

98 views

### Triviality of local system extension

Take a nice space $X$. Let us call a local system on $X$ a functor from the fundamental groupoid of $X$ to groups, so that $G$ is a local system on $X$ if for each $x \in X$ there is a group $G_x$ and ...

**8**

votes

**1**answer

578 views

### What is the analogue of a Lefschetz Thimble for Morse-Bott critical components (sets of non-isolated critical points)?

Small pre-face: I did an applied math PhD in the UK, but the problem I ended up studying has important ramifications in pure math, specifically to do with the Gauss-Manin connection in the presence of ...

**1**

vote

**0**answers

109 views

### A question about a manifold in an $n$-dimensional Alexandrov space with curvature bounded below [duplicate]

Suppose $M$ is an $n$-dimensional Alexandrov space with curvature bounded below(maybe with boundary), subspace $A\subset M$ is an $n$-dimensional manifold without boundary. Then whether every point in ...

**6**

votes

**1**answer

417 views

### homotopy fixed points and fixed points

Let $X$ a smooth projective scheme over a field $k$. And let $THH(X)$ denotes the topological Hochschild homology of $X$. Recall that the spectra $THH(X)$ admits an action of the of circle $S^{1}$. ...

**1**

vote

**1**answer

163 views

### Bounded dg algebra vs unbounded dg algebras

1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...

**1**

vote

**1**answer

136 views

### Lefschetz fixed notation

If $f\colon X\to X$ is a self-map of a nice space with isolated fixed points, then the Lefschetz fixed point theorem relates a global number to local numbers. Some write: $L(f)=\sum_{x\in ...

**23**

votes

**2**answers

726 views

### Unstable homotopy groups of spheres beyond Toda's range

In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are ...

**4**

votes

**1**answer

348 views

### Cap product à la Poincaré

Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...

**5**

votes

**0**answers

88 views

### Contractibility of a poset-indexed colimit

Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ ...

**1**

vote

**2**answers

166 views

### Generalization of Bracketing (or one of its many equivalences)

I asked the following question on MathStackExchange, but I have not received any answers after almost 3 days. Although it may not be a research level question, I thought I could ask it here.
*"Is ...

**13**

votes

**0**answers

538 views

### What is operator tmf?

One of the many wonderful things about K-theory, relative to other generalized cohomology theories, is that it can be defined for not-necessarily-commutative C*-algebras. The resulting construction, ...

**1**

vote

**0**answers

112 views

### Homotopical nilpotency of self homotopy equivalence

Given a topological space $X$, ${\rm aut}(X)$ denotes the monoid of the homotopy self equivalences of $X$, that are maps $f: X\rightarrow X$ which admits a homotopy inverse. ${\rm aut}_1(X)$ denotes ...

**3**

votes

**1**answer

219 views

### Reference request for non-commutative analogues of exterior algebras

I am reading Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (preprint available on web page). Here is an extract of the paper:
Cohen called $A^R_n$ "a standard tool used in ...

**9**

votes

**1**answer

399 views

### Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...

**2**

votes

**0**answers

164 views

### Dehn-Sommerville relations for $\Delta$-complexes

Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the ...

**1**

vote

**0**answers

117 views

### Question about a theorem of Goodwillie on periodic cyclic homology

In his paper Cyclic homology, Derivations and the Free Loopsace, Goodwillie defines periodic cyclic homology for differential graded algebras (A,d) concentrated in non-negative degree.
Why does he ...

**38**

votes

**2**answers

916 views

### Can a subset of the plane have nontrivial $H_2$ or $\pi_2$?

This is a question that occurred to me years ago when I was first learning algebraic topology. I've since learned that it's a somewhat aesthetically displeasing question, but I'm still curious about ...

**10**

votes

**1**answer

214 views

### String Orientation and Level Structures

Atiyah, Bott and Shapiro defined orientations of real and complex K-theory that were later refined to maps of ($E_\infty$-ring) spectra
$$MSpin \to KO$$
and
$$MSpin^c \to KU.$$
Likewise, but more ...

**0**

votes

**1**answer

159 views

### Principal bundle associated to a fiber bundle

Let $\pi : E\to B$ be a fiber bundle with fiber $F$ over a finite complex $B$ whose structure group is a compact Lie group $G$. How can we determine the principal $G$-bundle associated to $\pi$? For ...

**17**

votes

**1**answer

298 views

### Lens spaces and generalized Petersen graphs

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a ...

**0**

votes

**0**answers

55 views

### Lusternik-Schnirelmann category of rational spaces of infinite type

Suppose I have an $n$-dimensional (simply-connected) rational space $X$ with L-S category $k$. Must there be a finite rational subcomplex $K \subseteq X$ with $\mathrm{cat}(K) = k$?
Even better, I ...

**7**

votes

**1**answer

287 views

### Parity of self-linking

For a class $x$ in $H_{2k}$ of a 4k-manifold $M$, the self-intersection $x.x$ agrees mod 2 with the cap product of $x$ with the Wu class $v_{2k}$. If instead $x$ is a torsion element of $H_{2k}$ of a ...

**3**

votes

**1**answer

241 views

### Generalization of Giroux's Theorem for Higher Dimensions?

Just wanted to know if Giroux's theorem for 3-dimensional contact manifolds can be generalized:
In contact geometry for manifolds of dimension 3 , we have Giroux's theorem , stating that for any ...

**5**

votes

**1**answer

325 views

### Uniqueness of Complex Orientation of Morava K-theory

It is known that the $n^{\text{th}}$ Morava $K$-theory at a prime $p$, denoted $K(n)$, is complex oriented. In other words, it admits a theory of Chern classes, or equivalently a morphism of homotopy ...

**4**

votes

**2**answers

378 views

### $E_n$-space and n-connected pointed space

Is it true that the homotopy category of group-like $E_n$-spaces is equivalent to the homotopy category of pointed $n$-connected spaces ? If it is true, what should be the statement when ...

**6**

votes

**0**answers

107 views

### When is the diagonal inclusion a $\Sigma_2$-cofibration?

Recall that a space $X$ is called locally equiconnected or LEC if the diagonal map $d:X\hookrightarrow X\times X$ is a cofibration. For example, CW-complexes are LEC. There is some discussion of this ...

**2**

votes

**2**answers

239 views

### Closure relations between Bruhat cells on the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...

**0**

votes

**0**answers

91 views

### Does the polynomial De Rahm functor preserves finite cartesian products?

Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets.
I have the following questions
1) When we have a quasi-isomorphism between ...

**4**

votes

**1**answer

237 views

### Is the classifying space of a symmetric monoidal category an infinite loop space?

Wikipedia states:
The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an infinite loop space.
If my mind is correct, Segals delooping machine gives a ...

**3**

votes

**1**answer

80 views

### Local index formula for >ungraded< elliptic operators

Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic ...

**7**

votes

**0**answers

143 views

### Singular cohomology of $BG$ and Borel cohomology of $G$

Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner.
For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, ...

**3**

votes

**1**answer

260 views

### Question about the h-principle

So generally we define a differential relation to be $\mathcal{R} \subset X^{(r)}.$ In the case that $X=M\times N$ is it possible to have $\mathcal{R}=X^{(1)}$? So in this case the formal solutions ...

**2**

votes

**0**answers

128 views

### Is complex surface in CP(3) a two handlebody?

Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take
\begin{equation}
S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\}
...

**2**

votes

**0**answers

99 views

### Generalization of Jordan Curve Theorem

Jordan Curve Theorem says that any plane continuum homeomorphic to $\mathbb{S}^1$ separates the plane into exactly two components.
Now
"Let $\alpha$ and $\beta$ be two homeomorphic plane continua. ...

**7**

votes

**1**answer

292 views

### Thom isomorphism from the ABGHR perspective

In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is ...

**6**

votes

**1**answer

215 views

### Action of the homotopy braid groups on reduced free groups

Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...

**0**

votes

**0**answers

20 views

### Action as specified in the extension given by the holomorph

I am reading Braid Groups, Free Groups, and The Loop Space of the 2-sphere
by F.R. Cohen and J. Wu, which is available here.
Here is an extract of the paper:
(Please refer to the paper for ...

**7**

votes

**0**answers

381 views

### Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action

Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$.
Consider the $\infty$-category ...

**2**

votes

**0**answers

129 views

### Which homology classes from loop space?

Fix a closed connected manifold $Q$ and let $LQ$ denote its free loop space. We can get second homology classes on $Q$ by "doing things" to loops in $Q$.
For instance, if we have a loop of loops, it ...