Below you will find example sentences with "fundamental group". The examples show how this phrase is used in natural context and which words often surround it.

Fundamental Group in a sentence

Corpus data

  • Displayed example sentences: 17
  • Discovered as a combination around: fundamental
  • Corpus frequency in the collocation scan: 8
  • Phrase length: 2 words
  • Average sentence length: 23.9 words

Words in this phrase

fundamental group

Meanings of words in this phrase

  • fundamental can mean: A basic truth, elementary concept, principle, rule, or law. An individual fundamental will often serve as a building block used to form a... Frequent word
  • group can mean: A number of things or persons being in some relation to one another. Very common word

Sentence profile

  • Phrase position: 9 start, 7 middle, 1 end
  • Sentence types: 16 statements, 1 questions, 0 exclamations

Usage patterns

These patterns show short text windows around the phrase in the example sentences:

the fundamental group of the 2× also trivial fundamental group had to 1× as the fundamental group of a 1× compute the fundamental group of an 1× example fundamental group of torus 1× example the fundamental group of the 1× in the fundamental group of a 1× natural and fundamental group unit of 1×

Sentence openings and endings

Common sentence openings:

the fundamental group 4× example fundamental group 1× for example the 1× for instance the 1× furthermore the latter 1× geometric interpretation conjugacy 1×

Common sentence endings:

2 or higher 1× 3 dimensional sphere 1× a 3 sphere 1× arbitrary topological space 1× around the hole 1× been violated extensively 1×

Context words

These words often appear near this phrase in the example sentences:

space 4× topological 4× free 4× example 3× turns 3× trivial 3× sphere 3× torus 2× functorial 2× homotopy 2× case 2× homeomorphic 2×

Related phrases

group equities lie group advisory group ethnic group terror group terrorist group financial group group inc fundamental rights group incorporated

Phrases around this combination

Other phrases discovered around fundamental:

fundamental rights 12× fundamental human 11×

Just before it in the phrase index:

fund managers fund launched functioning properly functional programming function keys fully understand

Just after it in the phrase index:

fundamental human fundamental rights funding boost fundraising campaign fundraising efforts fundraising event

Other phrases with the same pattern

Same start:

fundamental rights fundamental human

Same ending:

lie group advisory group ethnic group terror group terrorist group financial group

Corpus analysis

  • The phrase "fundamental group" has 2 words and usually appears near the start in these examples. The average sentence has 23.9 words and is mostly made up of statements.
  • Around this phrase, patterns and context words such as the fundamental group of the, also trivial fundamental group had to, space, topological and free stand out.
  • In the phrase index, this combination connects with group equities, lie group, advisory group, fundamental rights and fundamental human, linking the page to nearby combinations.

Example types with fundamental group

This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:

Short example sentences

This way, the fundamental group detects the hole. (8 words)

The fundamental group functor takes products to products and coproducts to coproducts. (12 words)

The fundamental group is a topological invariant : homeomorphic topological spaces have the same fundamental group. (15 words)

Longer example sentences

The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just X. It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space. (40 words)

The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. (35 words)

For instance, the right to the protection of the family as the “natural and fundamental group unit of society”, guaranteed under the Universal Declaration of Human Rights of 1948, has been violated extensively. (33 words)

Question examples

Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere? (23 words)

Example sentences (17)

Example: fundamental group of torus As an example of the distinction between the functorial statement and individual objects, consider homotopy groups of a product space, specifically the fundamental group of the torus.

The condition on the fundamental group turns out to be necessary (and sufficient) for finite time extinction, and in particular includes the case of trivial fundamental group.

The fundamental group is a topological invariant : homeomorphic topological spaces have the same fundamental group.

The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators.

Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial.

For example, the fundamental group of the figure eight is the free group on two letters.

Realizability *Every group can be realized as the fundamental group of a connected CW-complex of dimension 2 (or higher).

The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just X. It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space.

Advertentie

The fundamental group of the plane punctured at n points is also the free group with n generators.

For instance, the right to the protection of the family as the “natural and fundamental group unit of society”, guaranteed under the Universal Declaration of Human Rights of 1948, has been violated extensively.

Furthermore, the latter formula is a special case of the Seifert–van Kampen theorem which states that the fundamental group functor takes pushouts along inclusions to pushouts.

Geometric interpretation Conjugacy classes in the fundamental group of a path-connected topological space can be thought of as equivalence classes of free loops under free homotopy.

In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere.

Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

The fundamental group functor takes products to products and coproducts to coproducts.

The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole).

This way, the fundamental group detects the hole.

Advertentie