On path-components of the mapping spaces $$M(\mathbb {S}^m,\mathbb {F}P^n)$$ M ( S m , F P n )

Golasiński, Marek; Melo, Thiago de; Santos, Edivaldo L. dos

doi:10.1007/s00229-018-1012-5

Cited by 6 publications

(8 citation statements)

References 21 publications

(31 reference statements)

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“…However, after posting to arXiv the initial version of the present paper Prof. Helge Glöckner kindly informed us that such a statement follows from his (currently unpublished) paper [14, Proposition 6.1], see Remark 7.4 for details. That weak homotopy equivalence is a consequence of C ∞ approximation theorems and extends [36,Theorems 12,16,17] and [48] about homotopy types of open subsets of locally convex topological vector spaces.…”

Section: Introductionsupporting

confidence: 58%

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Smooth approximations and their applications to homotopy types

Khokhliuk,

Maksymenko

2020

Preprint

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Let M, N the be smooth manifolds, C r (M, N ) the space of C r maps endowed with the corresponding weak Whitney topology, andIt is also established a parametrized variant of such a result. In particular, it is shown that for a compact manifold M , the inclusion of the space of C s isotopies η : [0, 1] × M → M fixed near {0, 1} × M into the space of loops Ω(D r (M ), id M ) of the group of C r diffeomorphisms of M at id M is a weak homotopy equivalence.

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Section: Introductionsupporting

confidence: 58%

“…A good survey on the homotopy types of spaces of smooth maps can be found in [42], see also [19,39,41,1,2,16], and [7,43,44,24,45] for the topology of loop spaces.…”

Section: Introductionmentioning

confidence: 99%

Smooth approximations and their applications to homotopy types

Khokhliuk,

Maksymenko

2020

Preprint

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“…However, after posting to arXiv the initial version of the present paper Prof. Helge Glöckner kindly informed us that such a statement follows from his (currently yet unpublished) paper [14,Proposition 6.1], see Remark 7.5 for details. That weak homotopy equivalence is a consequence of C 8 approximation theorems and extends [36,Theorems 12,16,17] and [49] The aim of this paper is to give elementary and thorough proofs that several kinds of natural inclusions between spaces of differentiable maps, including inclusions (1.1), are (weak) homotopy equivalences as well, see Theorems 1.5, 1.8, and 8.1.…”

Section: Introductionmentioning

confidence: 81%

“…A good survey on the homotopy types of spaces of smooth maps can be found in [43], see also [19,40,42,1,16], and [7,44,45,24,46] for the topology of loop spaces. Moreover, for a subset X Ă M and a continuous map φ : X Ñ N define the following subspaces of B and A with the induced topologies:…”

Section: Introductionmentioning

confidence: 99%

Smooth approximations and their applications to homotopy types

Khokhliuk

Maksymenko

2020

PIGC

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Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset.It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence.It is also established a parametrized variant of such a result.In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $\eta:[0,1]\times M \to M$ fixed near $\{0,1\}\times M$ into the space of loops $\Omega(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.

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“…Indeed, other techniques, such as median nerve stimulation (MNS) that target the sensory system, can have a significant impact on motor cortex plasticity (Lai et al, 2016). The cortical and subcortical effects of MNS are thought to be mediated by bottom-up mechanisms, namely by a mechanism of coactivation that can trigger activation of the primary and secondary somatosensory areas (Hodzic, 2004), as well as the insula (Ferretti et al, 2007;Hodzic, 2004;Ibanez et al, 1995), and other cortical areas (Golaszewski et al, 2004;Manita et al, 2015;Wu, Van Gelderen, Hanakawa, Yaseen, & Cohen, 2005). Moreover, it has already been shown that the motor cortex and the sensory cortex can be comodulated by peripheral electric stimulation (Schabrun, Ridding, Galea, Hodges, & Chipchase, 2012) and that this sensorimotor integration is crucial for the acquisition and performance of motor skills (Arce-McShane, Ross, Takahashi, Sessle, & Hatsopoulos, 2016).…”

mentioning

confidence: 99%

Median nerve stimulation induced motor learning in healthy adults: A study of timing of stimulation and type of learning

Carvalho

French

Thibaut

et al. 2018

Eur J of Neuroscience

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Median nerve stimulation (MNS) has been shown to change brain metaplasticity over the somatosensory networks, based on a bottom-up mechanism and may improve motor learning. This exploratory study aimed to test the effects of MNS on implicit and explicit motor learning as measured by the serial reaction time task (SRTT) using a double-blind, sham-controlled, randomized trial, in which participants were allocated to one of three groups: (a) online active MNS during acquisition, (b) offline active MNS during early consolidation and (c) sham MNS. SRTT was performed at baseline, during the training phase (acquisition period), and 30 min after training. We assessed the effects of MNS on explicit and implicit motor learning at the end of the training/acquisition period and at retest. The group receiving online MNS (during acquisition) showed a significantly higher learning index for the explicit sequences compared to the offline group (MNS during early consolidation) and the sham group. The offline group also showed a higher learning index as compared to sham. Additionally, participants receiving online MNS recalled the explicit sentence significantly more than the offline MNS and sham groups. MNS effects on motor learning have a specific effect on type of learning (explicit vs. implicit) and are dependent on timing of stimulation (during acquisition vs. early consolidation). More research is needed to understand and optimize the effects of peripheral electrical stimulation on motor learning. Taken together, our results show that MNS, especially when applied during the acquisition phase, is a promising tool to modulate motor leaning.

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On path-components of the mapping spaces $$M(\mathbb {S}^m,\mathbb {F}P^n)$$ M ( S m , F P n )

Cited by 6 publications

References 21 publications

Smooth approximations and their applications to homotopy types

Smooth approximations and their applications to homotopy types

Smooth approximations and their applications to homotopy types

Median nerve stimulation induced motor learning in healthy adults: A study of timing of stimulation and type of learning

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