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

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Splitting the Hopf map in two

Given the Hopf map $h:S^3\to S^2$ and an inclusion $i:S^2\hookrightarrow S^3$, the map $i\circ h:S^2\to S^2$ has mapping degree zero. Therefore, it is homotopic to the constant map and the image of ...
9
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2answers
512 views

How much of homotopy theory can be done using only finite topological spaces?

Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is ...
8
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2answers
295 views

Fundamental group of $\mathbb{R}^3-F$ where $F\subseteq \mathbb{R}\times \{0\} \times \{0\}$

Maybe not research level. Let $Z\cong \mathbb{R}$ be the $z$-axis of $\mathbb{R}^3$. Clearly $\pi_1(\mathbb{R}^3-Z)\cong \mathbb{Z}$. Now if $F\subset Z$ is a closed non-empty subset, then one easily ...
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2answers
158 views

If an abelian category $\mathcal{A}$ has enough injectives then so does $\mathrm{Ch}^{\geq 0}(\mathcal{A})$

Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if ...
4
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1answer
316 views

Connections between Standard, Hodge and Tate conjectures on algebraic cycles?

What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
4
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68 views

Connection between framed cobordisms and zero sets

Let $W\subseteq M\times[0,1]$ be a framed submanifold (a framed cobordism in $M$) and $2w<m-1$ where $w,m=\dim W,M$. Assume that $M$ is compact and that $W\cap M\times \{0\}$ is the zero set of a ...
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83 views

Could anyone help me with a problem regarding fundamental groups? [on hold]

Let G be a group and x be an element of G. N is the least normal subgroup of G containing x. If there is a normal, path-connected space whose fundamental group is isomorphic to G, then I have to show ...
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157 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 ...
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0answers
126 views

On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces, $$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
13
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4answers
338 views

Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is: "For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
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238 views

Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
2
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1answer
203 views

RO(G) grading of Mackey functors

If G is a finite group, I understand that the category of RO(G)-graded spectra, when rationalized, becomes Quillen equivalent to the category of Mackey functors valued in chain complexes of rational ...
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164 views

Research topic selection [closed]

I am planning to do my PhD in either sheaf theory or generalised functions.pls help me choose one among them by pointing the scope and relevance. thanks in advance.
8
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1answer
237 views

Representation of finite groups in a compact Lie group

Let $H$ be a finite $p$-group, and let $G$ be a compact connected Lie group. Then it is well-known that $[BH,BG]\cong Rep(H,G)$, where $BH$ and $BG$ are classifying spaces and $Rep(H,G)$ is the set ...
3
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1answer
152 views

Carlsson's spectrum BG^-V

In the appendix to Carlsson's "Equivariant stable homotopy and Segal's Burnside ring conjecture," he introduces a spectrum BG^-V associated to a G-representation V. It is like a Thom spectrum of the ...
4
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2answers
263 views

$K$-homology of $BG$

Let $G$ be a finite group. Atiyah proved that the $K$-cohomology of $BG$ vanishes in odd degrees and in even degrees is the completion of the representation ring of $G$ at the augmentation ideal. ...
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65 views

Video Lecture for topolgy [closed]

want to study general topology. i find video lecture at nptel, but instructor's pronunciation is hard to understand. so other course to study general topology? thankyou!
4
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1answer
142 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 ...
11
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2answers
278 views

The definition of Reedy category

The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its ...
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0answers
247 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?
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0answers
79 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. ...
1
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1answer
137 views

Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X \cong \oplus_a ...
2
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3answers
446 views

Reference request: SGA7

I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?
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129 views

Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
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72 views

Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates ...
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1answer
262 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 ...
2
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0answers
47 views

d-refining covering of normal space

If $X$ is normal, it is well known that for any open-covering $(U_i)$ of $X$, there exist closed subspaces $F_i$ and $G_i$ and an open subspaces $O_i$ such that $$F_i\subset O_i\subset G_i\subset ...
4
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1answer
224 views

Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$: $0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
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0answers
415 views

A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
5
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2answers
274 views

Time-line until the publicaton of Weil of “Numbers of solutions of equations in finite fields”

In "On the history of the Weil Conjectures" Dieudonné says: "Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...". C. F. Gauss, Disquisitiones ...
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3answers
216 views

question about the induced homomorphism of etale fundamental groups

Background/Setup For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...
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Homotopy type of complex algebraic varieties

In his 1974 ICM adress "Poids dans la cohomologie des variétés algébriques", Pierre Deligne explains that any finite polyhedron has the same homotopy type as a complex algebraic variety (section 6.). ...
17
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2answers
380 views

Adams Operations on $K$-theory and $R(G)$

One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...
6
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2answers
132 views

Topological relationships between family of transversal intersections of manifolds

Let $M$ and $N$ be submanifolds of $\mathbb{R}^n$ and let $a(t)$ be a smooth path in $\mathbb{R}^n$ such that $M+a(t)$ intersects $N$ transversally for all $t \in [0,1]$. Is there a nice relationship ...
4
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2answers
222 views

A question about Dehn surgery and Brieskorn homology 3-spheres

I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres." If I ...
5
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2answers
194 views

What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$

Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. ...
2
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1answer
179 views

Simple connectedness of $\mathbb{C}P^2$ intersected with an affine subspace

The complex projective plane $\mathbb{C}P^2$ can be thought of as the set of rank one 3-by-3 Hermitian matrices with norm one, i.e., $\mathbb{C}P^2 = \{xx^* : x \in \mathbb{C}^3, x^*x=1 \}$. As such, ...
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71 views

generalisation of the universal coefficient spectral sequence

Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective ...
6
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1answer
267 views

Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case?

Suppose that $\mathcal{M}$ is a model category which is not combinatorial, does a homotopy limit in $\mathcal{M}$ correspond to a limit in the associated $\left(\infty,1\right)$-category? How about ...
4
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1answer
238 views

A question on fixed point theory

I asked this question in MSE, but I did not received any answer, so I repeat it here: http://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property Assume that $0<k<n-1$, ...
15
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4answers
455 views

Multiplicative Structures on Moore Spectra

The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
3
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3answers
755 views

Why are we interested in the Fundamental Groupoid of a Space?

The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$. ...
6
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107 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 ...
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2answers
209 views

Morphisms every pushout of which is a weak equivalence

Let $M$ be a category equipped with a class of weak equivalences $W$. Is there a name for a morphism $f$ such that every pushout of $f$ (including, of course, $f$ itself) is a weak equivalence? For ...
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Covering Spaces and Vector Bundles

Suppose $f: Y \rightarrow X$ is a covering map between compact Hausdorff spaces $X$ and $Y$. Then $f$ induces a algebra homomorphism $f^*:C(X) \rightarrow C(Y)$ and gives $C(Y)$ the structure of a ...
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246 views

From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
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9answers
2k views

Why localize spaces with respect to homology?

A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...
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94 views

Does simplicial localization with a 3-arrow calculus commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...
2
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1answer
462 views

1D TQFT in Freed-Hopkins-Lurie-Teleman

In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory. $F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual. $F(\circ-\circ)$ ...
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2answers
390 views

What does Freed-Hopkins-Teleman say about finite groups?

The third in the "Loop groups and twisted K-theory" series by FHT treats compact Lie groups without any connectedness assumptions. I am trying to unwind what Theorem 2 of that paper (available here, ...