Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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6 votes
0 answers
217 views

Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space

In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background "Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...
1 vote
1 answer
285 views

What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure?

Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion ...
5 votes
1 answer
321 views

K-theory of finite diagram categories

Suppose $I$ is a finite $\infty$-category and $F:I\rightarrow\text{fCW}$ is a functor that takes values in finite CW complexes. For each $X\in I$, let $[F(X)]$ denote the class of $F(X)$ in $K_0(\text{...
8 votes
0 answers
125 views

Relating bordism generators in d and d+2 dimensions --- an explicit example

This is an attempt to make my relation between bordism invariants in $d$ and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
2 votes
1 answer
141 views

Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group

Let $M^d$ be a nontrivial bordism generator for the bordism group $$ \Omega_d^G= \mathbb{Z}_n, $$ suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...
8 votes
0 answers
191 views

"Gerbes" in the cobordism theory

In a lecture I attended today, I heard the use of gerbes in the cobordism theory. Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group ...
5 votes
1 answer
270 views

Manifold generators of O-bordism invariants

If I understand correctly, I can obtain the $O$-cobordism group of $$ \Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4, $$ The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...
6 votes
1 answer
142 views

Example of nonvanishing Waldhausen Nil group

In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen ...
5 votes
1 answer
189 views

Relating bordism groups of different dimensions

Let $M_d$ be a $d$-manifold generator of a subgroup of bordism group $$ \Omega_d^{G}, $$ or further generalization $$ \Omega_d^{G}(K(\mathcal{G},n+1)), $$ which $G$ is the given structure ...
1 vote
0 answers
143 views

Example of open manifold with no free integer homology non-homeomorphic to a ball

I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball. ...
2 votes
0 answers
71 views

Topological Shape Operator More Sensitive than Inverse Limits

This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
4 votes
0 answers
279 views

Generalized Postnikov square

Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
0 votes
1 answer
366 views

How to prove that there does not exist any plane isotopy from the logarithmic spiral onto the real line? [closed]

Questions. EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a strong a-forteriori-reason, kindly ...
4 votes
1 answer
381 views

null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes

$\require{AMScd}$ Preliminaries: Let $\Sigma$ be a closed manifold, $X$ be a CW complex and $f:\Sigma \to X$ be a map. We say that the pair $(\Sigma,f)$ is null-homologous (over $\mathbb{Z}_2$) if $...
9 votes
2 answers
582 views

Künneth formulas/theorem for bordism groups and cobordisms?

We are familiar with Künneth theorem: The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...
5 votes
1 answer
207 views

Cellularity of anodyne extensions?

Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts) If not, is there a known counterexample? Similarly, does ...
4 votes
1 answer
793 views

homologies of some subsets of ${R}^{n}$

This might be something well-known. For $1\le k\le n$, let $A(n,k)\subset\mathbb{R}^{n}$ be the set of points $x=(x_{1},...,x_{n}% )$ with at least $k$ distinct coordinates. Then what are the ...
8 votes
1 answer
432 views

Spin cobordism v.s. KO theory in low or in any dimensions

It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension. If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$ ...
1 vote
1 answer
385 views

Formality of surfaces

The de Rham dg algebra $\Omega(F)$ of a closed orientable surface $F$ is formal (that is, weakly equivalent to its cohomology algebra). This is a special case of the fact of formality of Kähler ...
6 votes
1 answer
189 views

The Stone-Čech compactification of the fixed point set

Let $G$ be a discrete group and $X$ be a Tychonoff $G$-space. Then there exists a $G$-action on Stone-Čech compactification $\beta X$. If the fixed point set $X^{G}\neq \emptyset $, then the Stone-...
3 votes
1 answer
138 views

Bordism invariants vanishes in a lifted twisted $Pin^- \times Spin$-structure

It looks to me that the bordism group $$\Omega_3^{SO} (B(O(2) \times SO(3))) \tag{1}$$ (whose Pontryagin dual for the manifold generator) contains at least a nontrivial invariant: $$ w_1(O(2))\big(...
34 votes
3 answers
8k views

Different way to view action of fundamental group on higher homotopy groups

There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at ...
4 votes
1 answer
349 views

(Co)bordism invariant of Eilenberg–MacLane space becomes vanished

Consider a (co)bordism invariant $$ u_2 Sq^1 u_2+Sq^2 Sq^1 u_2 $$ obtained from $$ \Omega^5_{O}(K(\mathbb{Z}/2,2)). $$ Here $u \in H^2(K(\mathbb{Z}/2,2),\mathbb{Z}_2)$. The $K(\mathbb{Z}/2,2)$ is ...
6 votes
1 answer
289 views

HKR generalized character theory question regarding a certain construction

In that paper https://web.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf Hopkins-Kuhn-Ravenel introduced the idea of generalized character corresponding to a complex-oriented cohomology ...
5 votes
2 answers
261 views

Naturality of PD model of a CDGA

In the paper "Poincaré duality and commutative differential graded algebras", Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are ...
2 votes
0 answers
67 views

Order relation between cohomology groups

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex $$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
4 votes
0 answers
165 views

Complex projective algebraic variety, moduli space of flat connections, and instantons

In Looijenga's work below, if I understand correctly, it shows that Statement 1: At an algebraic variety, the moduli space of SU($N$) flat connections on a 2-torus $T^2$ is given by the space of ...
5 votes
0 answers
240 views

Homotopy functor calculus vs functor calculus in additive categories

Consider the Goodwillie calculus of a homotopy functor $F : \mathrm{Sp} \to \mathrm{Sp}$, where $\mathrm{Sp}$ denotes an appropriate model for spectra (... orthogonal spectra for instance). Then ...
5 votes
0 answers
67 views

Does the theorem that genera vanishing on even-dim complex projective bundles are elliptic also apply for integral-valued genera?

Ochanine proved in this paper that for genera taking values in $\mathbb{Q}$-algebras, vanishing on even-dimensional projective bundles is equivalent to being an elliptic genus (i.e. a specialization ...
7 votes
1 answer
382 views

Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
5 votes
0 answers
268 views

Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions. Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...
3 votes
0 answers
90 views

Cohen's definition of loop operations

I have a problem with understanding Cohen's definition, as written in "The Homology of Iterated Loop Spaces" by Cohen, Lada and May (Part III, chapter 5, at the end of chapter). So in order to define ...
12 votes
2 answers
606 views

Is there any significance to Bousfield localization in the non-derived context?

The term "Bousfield localization" of a category $C$ is used in roughly two different ways: There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...
7 votes
0 answers
224 views

The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons

In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
5 votes
1 answer
235 views

Pontryagin square, Postnikov square and their consistency formulas

$\mathcal{P}_2$ is Pontryagin square $$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$ $\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$ question (i)...
4 votes
1 answer
255 views

Trivialization of Pontryagin square on oriented $4$-manifolds

I'm sorry for not clearly stating my question, thanks to Robert Bruner for answering my original question, let me restate it. Let $\mathcal{P}:H^2(-,\mathbb{Z}/2)\to H^4(-,\mathbb{Z}/4)$ be the ...
8 votes
2 answers
683 views

Adem relations of Steenrod square without modding out the coboundaries

In the paper Products of Cocycles and Extensions of Mappings, Steenrod introduced the cup-$i$ product and Steenrod square $Sq^k$: $$ Sq^k(x_n) \equiv x_n \smile_{n-k} x_n,\ \ \ x_n \in C^n(M^d;\...
3 votes
1 answer
630 views

Bockstein homomorphism and Square Operations: Their consistency formulas

Here are various ways to define "Bockstein homomorphism:" Let $\beta_p:H^*(-,\mathbb{Z}_p) \to H^{*+1}(-,\mathbb{Z}_p)$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}...
3 votes
0 answers
173 views

Pontryagin square on spin and non-spin manifold

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
11 votes
3 answers
957 views

Waldhausen $K$-theory for $G$-spaces

I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it? Let $G$ be a finite group. We know that ...
3 votes
2 answers
509 views

Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$

Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
8 votes
1 answer
734 views

Modules over Hopf Algebras and $E_2$-algebras

Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf I was wondering if anybody knows of a nice relationship between ...
5 votes
1 answer
494 views

Computation of $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$ and Smith isomorphism

question: I am looking for the literature with the result or the computation of Pin- bordism group: $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$. Can someone point out some useful ways to do this or any helpful ...
31 votes
1 answer
2k views

Stable homotopy theory and physics

At various points in my life, I have held the following beliefs: 1) Stable homotopy theory is "easy" rationally, and "interesting" integrally. 2) The spectrum of topological modular forms (TMF) is ...
10 votes
2 answers
2k views

Sphere spectrum, Character dual and Anderson dual

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. However, could you help me to appreciate the mathematical meanings of the following: What is the significance of ...
18 votes
1 answer
2k views

Homology theory represented by Madsen-Tillmann spectra

The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO_n$ is bordism theory:\begin{equation*}\pi_k(MO\wedge X)=\Omega^O_k(X)\end{equation*}These groups form the ring of (...
7 votes
0 answers
221 views

Does Deligne's exceptional series lead to an "exceptional K-theory"?

To a certain extent, Deligne's exceptional series $A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$ plays a role analogous to the classical series $A_n \subset ...
5 votes
2 answers
3k views

How to prove that $w_1(E)=w_1(\det E)$?

How to prove that the first Stiefel-Whitney class $w_1 (E)$ of a real rank $n$ vector bundle over a manifold M is equal to $w_1(\det E)$, where $\det E$ is the $n$-th wedge power of $E$? (I want to ...
2 votes
0 answers
92 views

Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

We know well this short exact sequence $$ 1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1. $$ The $j$-th Stiefel-Whitney class of the associated vector bundle of $SO(N)$, as $w_j(V_{SO(N)})$, can be ...
7 votes
1 answer
1k views

What's special about elliptic cohomology?

Apologies for any basic mistakes in this question; I'm a beginner to this theory and don't have anyone at my institution to consult for advice. What I mean is, if you take an elliptic curve $E$ over $...

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