Questions tagged [mapping-space]

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Finding the set of best approximation

Given $X$=$l^1$ and its dual space $X^*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^*$. Then clearly $\|f\|_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$. Note: A ...
0 votes
0 answers
167 views

Mapping between pseudosphere and breather segments

The pseudospheres $(K=-1)$ shown contain two cuspidal edge boundaries and appear to admit isometric mapping between 1) the surface of revolution through an expansion/contraction deformation in ...
0 votes
0 answers
78 views

Topologies in $\mathcal{C}^\infty(M,N)$

Naively, one could topologise the set of smooth (ie $\mathcal{C}^\infty$) maps between two smooth manifolds $M \to N$ with the subspace topology $\mathcal{C}^\infty(M,N) \subseteq \mathcal{C}^0(M,N)$, ...
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2 votes
0 answers
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Quantifierisation of maps

I will rewrite my question using Matt F. suggestion. Consider the logical structure $L = (\mathbb{R}, +, *, 1, 0, =)$ and a function $f:\mathbb{R}\to\mathbb{R}$. Consider the map $Q:2^\mathbb{R}→2^\...
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3 votes
0 answers
174 views

Confused about the definition of the Kahn-Priddy map

The Kahn-Priddy map is defined in various papers as follows: Define $i:\mathbb {RP}^{n-1}\rightarrow O(n)$ by taking $\ell\in \mathbb {RP}^{n-1}$ to the reflection across $\ell^\perp.$ Define $j:O(n)\...
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7 votes
1 answer
375 views

Does the continuous mapping space between topological manifolds always admit a Banach manifold structure?

Let $M$ and $N$ be smooth, i.e. $C^\infty$, manifolds. Suppose that $M$ is compact. Then for every $k \geq 0$ it is well known that $$C^k(M,N)$$ admits the structure of a smooth Banach manifold. I am ...
0 votes
0 answers
75 views

Can we construct a surjective mapping from $\mathbb{R}^{?}$ to this space?

(Note : I'm not sure about the tags, please re-tag this if you think you have the right tag). I am optimising a certain function over a certain space (that i will describe), and to use non-constraint ...
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3 votes
1 answer
314 views

Conformal map onto a circle, from an identification space composed of five squares

I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at ...
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3 votes
0 answers
63 views

Stability for mapping class groups, spaces of sections, and polynomial coefficient systems

Let $X$ be a simply connected space. Cohen and Madsen https://arxiv.org/abs/math/0601750 proved that the functor sending a surface with boundary M to $H_i(Map(M,X))$ has polynomial degree $\leq i$. ...
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2 votes
1 answer
165 views

Do submersions induce open maps between spaces of differentiable maps?

Let $X$, $Y$ and $Z$ be smooth manifolds. Any differentiable map $f \colon Y \rightarrow Z$ induces a continuous map $f_{\ast} \colon C^{\infty}(X, Y) \rightarrow C^{\infty}(X, Z)$ via composition $g \...
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2 votes
0 answers
52 views

Simplicial models for mapping spaces of filtered maps

Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan. Suppose that ${...
-1 votes
1 answer
82 views

Hierarchies of Operator Norms [closed]

Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \...
3 votes
1 answer
223 views

Morphisms of parametrized ring spectra

This is a follow-up to this question, in which Denis Nardin nicely explained that $$ \operatorname{Map}_{\operatorname{Fun}(X, \operatorname{Sp})}(E_X, E'_X) \simeq \operatorname{Map}(X, \operatorname{...
2 votes
1 answer
161 views

Are morphisms of parametrized spectra themselves parametrized morphisms of spectra?

Let $X$ be a fixed parametrizing space. Let $E$ and $E'$ be two spectra and let $E_X$ and $E'_X$ be their trivial parametrized versions. Intuitively I imagine that the morphisms of parametrized ...
0 votes
1 answer
3k views

Is the Haversine Formula or the Vincenty's Formula better for calculating distance?

Which is better for calculating the distance between two latitude/longitude points, The Haversine Formula or The Vincenty's Formula? Why? The distance is obviously being calculated on Earth. Does ...
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6 votes
1 answer
240 views

Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map $f:\mathbb{C}P^2\...
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4 votes
1 answer
128 views

Homotopy equivalence of maps with compact support and maps which vanish at infinity

Let $X$ be a nice space, maybe a manifold, and let $Y$ be a based space. What sort of conditions must we impose on $Y$ (and $X$ if need be) to get a homotopy equivalence $$ \mathcal C_c(X,Y) \simeq \...
8 votes
1 answer
727 views

cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, the homology Hopf algebra (1) $$ H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p) $$ for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...
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2 votes
0 answers
186 views

cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring $$ H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p) $$ is obtained for any ...
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2 votes
1 answer
564 views

When is the inclusion of a relative mapping space into a mapping space a cofibration?

Let $(X,A)$ and $(Y,B)$ be pairs of spaces and subspaces, let $\operatorname{Map}(X,Y)$ the space of maps $f:X\to Y$ equipped with the compact-open topology and let $\operatorname{Map}(X,A;Y,B)$ be ...
17 votes
1 answer
1k views

Whitehead product with identity on homotopy groups of spheres

For $n\geq 2$ let $(S^n,p)$ be the $n$-sphere with a base point $p$. Let $1:S^n\to S^n$ denote the identity map. Let us define the map $Wh_1: \pi_i(S^n,p)\to \pi_{i+n-1}(S^n,p), \alpha\mapsto [\...
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11 votes
3 answers
1k views

Induced map on path manifolds: is it a submersion?

Consider the following claim: Let $p:M \to N$ be a (surjective) submersion of finite-dimensional smooth manifolds. Let $J$ denote one of $[0,1],\ [0,1),\ (0,1]$. Then $p_*:M^J \to N^J$ is a ...
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0 votes
1 answer
761 views

Remap FFT frequency bin distribution

I've coded up the FFT for a dataset I'm working with. My intent is to create a waterfall plot of the result, but the problem I'm running into is if I change my input data size, then I get a different ...
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8 votes
1 answer
2k views

Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following: 1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
4 votes
1 answer
189 views

Contractible space of maps between Eilenberg-Mac Lane spaces, 2

Let $G$ and $H$ be torsion abelian groups. Are the following are equivalent: $\mathrm{Hom}(G, H) = 0$ $\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$ ? Clearly (2) implies (1).
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4 votes
1 answer
424 views

Contractible space of maps between Eilenberg-Mac Lane spaces

Suppose $A$ is an abelian torsion group, with no elements of order $p$, and let $P$ be an abelian $p$-group (i.e., the order of each element is a power of $p$). It sure seems to me that $$ \mathrm{...
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6 votes
0 answers
351 views

The Space of Cellular Maps

Let $X$ and $Y$ be CW complexes. Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
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