5
votes
2answers
221 views

Using Discrete Morse Theory to represent hom classes

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...
7
votes
1answer
209 views

Higher coherent multiplicative structures on S-algebras

In their book, Elmendorf, Kriz, May and Mandell describe a useful category of spectra, called S-modules, where S is the sphere spectrum. Ring objects in this category can be identified with spectra ...
10
votes
2answers
384 views

“Economic” CW-structure for Eilenberg-MacLane spaces?

The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even ...
4
votes
1answer
244 views

Construction of generalized Eilenberg-MacLane spaces

The Eilenberg-MacLane spaces $K(G,q)$ are readily generalized to study cohomology with local coefficients.The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are spaces with only two nnvanishing ...
3
votes
0answers
116 views

Flat Connections on the Cotangent Complex

I'm trying to find a reference which defines and discusses some properties of connections and flat connections on the cotangent complex in a homotopical setting. That is to say, a connection or flat ...
4
votes
1answer
128 views

In a Simplicial group the Degenerate elements don't intersect the kernel of the face maps.

Let $G_{\bullet}$ be a simplical group. I denote by $D_n \subset G_n$ the subgroup generated by all the degeneracy maps $s_i:G_{n-1} \to G_n$. I also denote $$M_n = \bigcap_{i>0} \mathrm{Ker} d_i ...
2
votes
1answer
261 views

The topology of the classifying space of U(n)

In $\textit{Atiyah and Bott: The Yang-Mills equations over riemann surfaces}$, there is a sentence about the topology of BU(n) (i.e. the classifying space of U(n)): Here $K(\mathbb{Z},n)$ means the ...
3
votes
0answers
136 views

Homotopy type of a locally contractible compact

Does a locally contractible compact space have the homotopy type of a finite CW complex? (I think it probably does, but I need a reference anyway.) EDIT: My intuition was wrong [to see why, read ...
5
votes
1answer
270 views

Morava $K(n)$'s are not $E_{\infty}$

I am looking for a reference/proof that shows that the Morava $K$-theory spectra, $K(n)$ are not $E_{\infty}$ ring spectra. I suspect that this should be a calculation but I can't quite get it right. ...
7
votes
1answer
446 views

Learning roadmap to TQFT from a mathematics perspective

I had asked a question on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posting it below. ...
3
votes
2answers
345 views

Zigzags and contractibility of categories

Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion: If there is ...
0
votes
0answers
173 views

K-theory of $\mathbb{RP}^\infty$

can anyone give some reference of K-theory and K-homology of $\mathbb{RP}^\infty$, both $K_0$ and $K_1$. PS: also posted in stackexchange
5
votes
0answers
216 views

Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let $q: D^k \to D^r$ be a map and $r \leq k$. Let $$W = S^k \sqcup D^r/\sim$$ where ...
4
votes
1answer
151 views

$RO(G)$-Graded Cohomology Theories

Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter ...
1
vote
0answers
262 views

Grothendieck's letter to serre

Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?
1
vote
0answers
88 views

A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof? Statement. ...
2
votes
3answers
471 views

Reference request: SGA7

I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?
5
votes
1answer
290 views

Detection of stable homotopy by K-theory spectra

This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from ...
6
votes
0answers
110 views

A Property of Generalized Equivariant Cohomology

Let $G_i$ be a compact Lie group, $i=1,2$, and let $E_{G_i}^*$ be a $\mathbb{Z}$-graded complex-oriented $G_i$-equivariant generalized cohomology theory with commutative products. Let $X_i$ be a ...
4
votes
0answers
134 views

Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here. Is there ...
4
votes
1answer
136 views

Homotopy groups of linearization of a space

If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural ...
6
votes
1answer
227 views

Group actions in a homotopy category

Let $M$ be a model category and $G$ a finite group, and equip the category $M^G$ of $G$-objects in $M$ with, say, a projective model structure. Then there is a canonical functor $$\mathrm{Ho}(M^G) ...
4
votes
0answers
173 views

Which ring spectra have some kind of exponential map turning addition into multiplication?

This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time. What will follow is sort of vernacular but whether it can be ...
8
votes
1answer
223 views

Shriek push-forward for parameterized spectra

In May and Sigurdsson's Parameterized Homotopy Theory, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow ...
5
votes
1answer
296 views

Spectral Sequences reference

What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep. I'm ...
7
votes
1answer
229 views

How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map $$ \mu: G\times ...
8
votes
2answers
276 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
3
votes
0answers
112 views

Infinity category of functors from a relative category to a model category

Let $M$ be a model category (maybe cofibrantly generated/combinatorial). Let $(C,W)$ be a relative category. I write $M^{(C,W)}$ for the full subcategory of $M^C$ on relative functors. This is a ...
8
votes
3answers
932 views

Is there a homology theory that gives a *necessary and sufficient* condition for homotopy equivalence?

Is there a (non-Abelian) homology theory that realizes the following: Let $X,Y$ be manifolds with complexes $C(X),C(Y)$. Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ ...
0
votes
0answers
119 views

Where are there defined objects between gerbes and bundle gerbes?

Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorphism. Does this exist in the literature?
13
votes
2answers
808 views

Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman: A connected open subset $D$ of the plane $\mathbb C$ is simply connected if and only if its complement $\widetilde D = \mathbb C \setminus D$ ...
8
votes
1answer
211 views

worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ...
11
votes
2answers
444 views

Postnikov's algebraic reconstruction of cohomology from homotopy invariants

In his short paper (1951) and longer monograph (1955), Postnikov introduced what I believe are now called Postnikov systems or towers. It is my understanding that Postnikov systems have since then ...
7
votes
2answers
219 views

Cokernel of the stable J-homomorphism at odd primes

Where can one learn about odd-primary components of the cokernel of the stable J-homomorphism? According to wonderful Wikipedia article on Homotopy groups of spheres, the "hard" part of the stable ...
1
vote
0answers
84 views

Euler class and self-intersection number of a surface in a 4-manifold [duplicate]

In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that For a compact oriented surface $X$ in a 4-dimensional oriented ...
11
votes
0answers
365 views

Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains: $$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$ where the right hand side denotes the free abelian ...
9
votes
0answers
256 views

Cohomology and impossible figures

In connection with the MO question Occurrences of (co)homology in other disciplines and/or nature I recalled Roger Penrose's “On the cohomology of impossible figures": ...
2
votes
0answers
165 views

Stacks and Maurer-Cartan elements

One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$. For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
4
votes
1answer
263 views

What's a good reference for the following obstruction theory yoga?

Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I ...
1
vote
0answers
122 views

fixed simplicial complex under group action

I have found in an article dealing with combinatorial manifolds the following definition: Let $C$ be a finite simplicial complex, and let $G$ be a finite group acting by automorphisms of $C$. The ...
5
votes
1answer
233 views

Image of J splitting

The image of J space is defined for $p$ odd as the homotopy fibre $J_{(p)}$ of the self map $$\psi^k - 1: BU_{(p)} \to BU_{(p)}.$$ Here, $\psi^k$ is an Adams operation, and $k \in \mathbb{N}$ ...
4
votes
0answers
78 views

Computing the Product Structure in Equivariant Cohomology via a Stratification and Thom-Gysin

Let $X$ be a smooth complex projective variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite $T$-invariant stratification of $X$ into smooth ...
3
votes
0answers
92 views

Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...
6
votes
1answer
186 views

Stiefel-Whitney classes of virtual vector bundles

Let $E=\xi-\eta$ be a virtual vector bundle over a compact base $B$, which we may assume is a CW complex. A quick and dirty way to define the total Stiefel-Whitney class $w(E)\in ...
11
votes
1answer
246 views

Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups $$ C^k = \{f: G^k \to A\} $$ and the coboundary map $$ \delta : C^k \to C^{k+1} $$ $$ (\delta f)(g_1, \ldots, g_{k+1}) ...
2
votes
1answer
223 views

Massey product in Dual Steenrod Algebra

Let $\tau_0$ be the element of dual Steenrod algebra $A_p^{*}$ at a prime $p$ which is dual to Bockstein $\beta \in A_p$. It is well known $\tau_0^2 =0$. Is it true/known that the elemnet $\xi_1$ ...
6
votes
2answers
493 views

All mapping space between CW complexes is a CW complex?

Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$. If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex? Can we know the cell structure of ...
3
votes
1answer
267 views

What is the “higher version” of chain homotopy in singular homology?

In basic algebraic topology, we know the following well-known chain homotopy theorem: Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...
7
votes
2answers
402 views

Equivariant Stratifications of a Variety

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are ...
1
vote
0answers
98 views

Is the $K$-Theory of $\mathbb{P}X$ a $\lambda$-Ring?

If we let $\mathbb{P}X$ denote the free commutative algebra generated by the spectrum $X$, then we have a weak equivalence $$ \mathbb{P}X\simeq \bigvee_r E\Sigma_{r+}\wedge_{\Sigma_r}X^{\wedge r}. $$ ...