대수적 위상수학

대수적 위상수학
학문명대수적 위상수학

대수적 위상수학(代數的位相數學, 영어: algebraic topology)은 추상대수학적 도구를 사용하여 위상 공간과 다양체들을 다루는 위상수학의 분야다.

대수적 위상수학의 도구

공간의 위상수학적 구조는 다음과 같은 대수적 구조로 나타낼 수 있다.

  • 호모토피와 호모토피 군은 두 위상 공간 사이의 연속 함수를 연속적으로 변형하는 과정을 나타내는 대상이다. 호모토피는 위상 구조보다 더 단순한 호모토피 구조만을 나타낸다. 고차 호모토피 군은 다루기 복잡하지만, 기본군이라고 불리는 1차 호모토피 군은 계산하기가 비교적 쉬우며 널리 쓰인다.
  • 호몰로지와 코호몰로지는 일련의 공리를 만족하는 아벨 군들이다. 호몰로지의 개념 자체는 매우 일반적이며, 위상수학 말고도 추상대수학이나 대수기하학에서도 쓰인다. 대수적 위상수학에서, (코)호몰로지는 위상 공간 속에 존재하는 고차원 "구멍"들을 나타낸다. 대수적 위상수학에서는 특이 호몰로지와 체흐 코호몰로지 등을 주로 사용한다.

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