Problem: Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. 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.
What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
The answer is the following: Mathematical objects


The French forces tried to subdue and capture the liberal government based in Saltillo. On September 21, 1864, José María Patoni and Jesús González Ortega lost against the French forces at the Battle of Estanzuelas; the supreme government led by President Juárez was forced to evacuate the city of Saltillo and relocate to Chihuahua. Juárez stopped in Ciudad Jiménez, Valle de Allende, and Hidalgo de Parral, in turn. He decreed Parral the capital of Mexico from October 2–5, 1864. Perceiving the threat from the advancing French forces, the president continued his evacuation through Santa Rosalía de Camargo, Santa Cruz de Rosales, and finally Chihuahua, Chihuahua. On October 12, 1864, the people of the state gave President Juárez an overwhelmingly supportive reception, led by Governor Ángel Trías. On October 15, 1864 the city of Chihuahua was declared the temporary capital of Mexico.
Which governor led the supportive reception of President Juarez?
Ángel Trías


Input: Modern history
The Qing dynasty (1644–1911) was founded after the fall of the Ming, the last Han Chinese dynasty, by the Manchus. The Manchus were formerly known as the Jurchens. When Beijing was captured by Li Zicheng's peasant rebels in 1644, the Chongzhen Emperor, the last Ming emperor, committed suicide. The Manchus then allied with former Ming general Wu Sangui and seized control of Beijing, which became the new capital of the Qing dynasty. The Mancus adopted the Confucian norms of traditional Chinese government in their rule of China proper. Schoppa, the editor of The Columbia Guide to Modern Chinese History argues, "A date around 1780 as the beginning of modern China is thus closer to what we know today as historical 'reality'. It also allows us to have a better baseline to understand the precipitous decline of the Chinese polity in the nineteenth and twentieth centuries."

During what time period did the Qing dynasty rule?
Output: 1644–1911


Input: Article: Certain genera of Gram-positive bacteria, such as Bacillus, Clostridium, Sporohalobacter, Anaerobacter, and Heliobacterium, can form highly resistant, dormant structures called endospores. In almost all cases, one endospore is formed and this is not a reproductive process, although Anaerobacter can make up to seven endospores in a single cell. Endospores have a central core of cytoplasm containing DNA and ribosomes surrounded by a cortex layer and protected by an impermeable and rigid coat. Dipicolinic acid is a chemical compound that composes 5% to 15% of the dry weight of bacterial spores. It is implicated as responsible for the heat resistance of the endospore.

Now answer this question: What are ribosomes in endospores are enclosed in?

Output: cortex layer


Article: "The most widespread form of interspecies bonding occurs between humans and dogs" and the keeping of dogs as companions, particularly by elites, has a long history. (As a possible example, at the Natufian culture site of Ain Mallaha in Israel, dated to 12,000 BC, the remains of an elderly human and a four-to-five-month-old puppy were found buried together). However, pet dog populations grew significantly after World War II as suburbanization increased. In the 1950s and 1960s, dogs were kept outside more often than they tend to be today (using the expression "in the doghouse" to describe exclusion from the group signifies the distance between the doghouse and the home) and were still primarily functional, acting as a guard, children's playmate, or walking companion. From the 1980s, there have been changes in the role of the pet dog, such as the increased role of dogs in the emotional support of their human guardians. People and dogs have become increasingly integrated and implicated in each other's lives, to the point where pet dogs actively shape the way a family and home are experienced.

Question: A grave from 12,000 BC was found to contain an older person and what else?
Ans: puppy


Input: British Empire
While the Suez Crisis caused British power in the Middle East to weaken, it did not collapse. Britain again deployed its armed forces to the region, intervening in Oman (1957), Jordan (1958) and Kuwait (1961), though on these occasions with American approval, as the new Prime Minister Harold Macmillan's foreign policy was to remain firmly aligned with the United States. Britain maintained a military presence in the Middle East for another decade. In January 1968, a few weeks after the devaluation of the pound, Prime Minister Harold Wilson and his Defence Secretary Denis Healey announced that British troops would be withdrawn from major military bases East of Suez, which included the ones in the Middle East, and primarily from Malaysia and Singapore. The British withdrew from Aden in 1967, Bahrain in 1971, and Maldives in 1976.

Who was Harold Wilson's Defense Secretary?
Output:
Denis Healey