I går var jag tvungen att stanna hemma efter en liten olycka på cykeln.

This is because Swedish follows the V2 rule in main clauses. That means the finite verb must come in the second position.

So:

• I går = first position
• var = second position
• jag = comes after the verb

Compare:

• Jag var tvungen att stanna hemma i går.
• I går var jag tvungen att stanna hemma.

Both are correct, but when i går comes first, the verb must come before the subject.

Vara tvungen att is a very common Swedish expression meaning to have to, to be forced to, or to be obliged to do something.

In this sentence:

• var = was
• tvungen = forced/obliged
• att stanna = to stay

So var jag tvungen att stanna means I had to stay.

It is an idiomatic way to express necessity in the past.

Because the tense is already carried by var.

In vara tvungen att + infinitive, the second verb stays in the infinitive:

• Jag är tvungen att stanna = I have to stay
• Jag var tvungen att stanna = I had to stay

If you said Jag stannade hemma, that would simply mean I stayed home.
But Jag var tvungen att stanna hemma adds the idea of necessity: I had to stay home.

Swedish makes an important distinction here:

• hem = movement toward home
• hemma = location at home

So:

• Jag åker hem = I am going home
• Jag är hemma = I am at home
• Jag stannar hemma = I stay at home

Since the sentence is about remaining at home, hemma is the correct form.

Because olycka is an en-word in Swedish.

That affects both the article and the adjective:

• en olycka = an accident
• en liten olycka = a small/little accident

The adjective liten is a bit irregular. Compare:

• en liten olycka
• ett litet barn
• små olyckor

So here liten agrees with the common-gender noun olycka.

Efter means after.

In this sentence, it shows that the accident happened first, and staying home happened afterward:

• efter en liten olycka = after a small accident

It can also suggest the reason for staying home, since the accident is what led to that situation.

Because på cykeln means on the bike, which describes the situation physically.

• på cykeln = on the bicycle
• med cykeln = by bike / using the bike

So:

• en olycka på cykeln = an accident while on the bike
• jag åkte med cykeln = I went by bike

Here the speaker is describing where the accident happened, so på cykeln is the natural choice.

Swedish often uses the definite form when the thing is already clear from the situation.

Here, cykeln means the bike involved — naturally understood as the one the speaker was riding. Swedish does this quite often where English might choose a possessive or leave things more general.

So:

• på cykeln = on the bike
• på min cykel = on my bike

Both are possible, but på cykeln sounds natural when ownership is obvious or unimportant.

Yes, you can hear that, especially in everyday speech when the time expression i går already makes the past clear.

But var tvungen att is often the clearer and more standard way to express past necessity:

• I går var jag tvungen att stanna hemma = Yesterday I had to stay home

So for learners, var tvungen att is a very useful pattern to know for talking about something you had to do in the past.